# Comparison of the Stellar Populations of Bulges and Discs using the MaNGA
Survey

Philip Lah  Research School of Astronomy and Astrophysics, Australian National
University, Canberra, ACT 2611, Australia  [ Nicholas Scott  Sydney Institute
for Astronomy, School of Physics, University of Sydney, NSW 2006, Australia
Tania M. Barone  Centre for Astrophysics and Supercomputing, Swinburne
University of Technology, P.O. Box 218, Hawthorn, VIC 3122, Australia  A. S.
G. Robotham  International Centre for Radio Astronomy Research, University of
Western Australia, 35 Stirling Highway, Crawley WA 6009, Australia  Francesco
D’Eugenio  Cavendish Laboratory and Kavli Institute for Cosmology, University
of Cambridge, Madingley Rise, Cambridge, CB3 0HA, United Kingdom  Matthew
Colless  Research School of Astronomy and Astrophysics, Australian National
University, Canberra, ACT 2611, Australia  Sarah Casura  Hamburger Sternwarte,
Universität Hamburg, Gojenbergsweg 112, D-21029 Hamburg, Germany

(dd Mmm YYYY; dd Mmm YYYY; dd Mmm YYYY)

###### Abstract

We use the MaNGA integral-field spectroscopic survey of low-redshift galaxies
to compare the stellar populations of the bulge and disc components,
identified from their Sérsic profiles, for various samples of galaxies. Bulge
dominated regions tend to be more metal-rich and have slightly older stellar
ages than their associated disc dominated regions. The metallicity difference
is consistent with the deeper gravitational potential in bulges relative to
discs, which allows bulges to retain more of the metals produced by stars. The
age difference is due to star formation persisting longer in discs relative to
bulges. Relative to galaxies with lower stellar masses, galaxies with higher
stellar masses tend to have bulge dominated regions that are more metal-rich
and older (in light-weighted measurements) than their disc dominated regions.
This suggests high-mass galaxies quench from the inside out, while lower-mass
galaxies quench across the whole galaxy simultaneously. Early-type galaxies
tend to have bulge dominated regions the same age as their disc dominated
regions, while late-type galaxies tend to have disc dominated regions
significantly younger than their bulge dominated regions. Central galaxies
tend to have a greater metallicity difference between their bulge dominated
regions and disc dominated regions than satellite galaxies at similar stellar
mass. This difference may be explained by central galaxies being subject to
mergers or extended gas accretion bringing new, lower-metallicity gas to the
disc, thereby reducing the average metallicity and age of the stars; quenching
of satellite discs may also play a role.

###### doi:

10.1017/pasa.2020.32

###### keywords:

galaxies: bulges; galaxies: structure; galaxies: stellar content

Philip<EMAIL_ADDRESS>ARC Centre of Excellence for All Sky
Astrophysics in 3 Dimensions (ASTRO 3D) ARC Centre of Excellence for All Sky
Astrophysics in 3 Dimensions (ASTRO 3D) dd Mmm YYYY

##  1 Introduction

Galaxies naturally split into two principal stellar components: a bulge and a
disc. In the Milky Way, the bulge is old ($>$10 Gyr) and metal-rich ([Z/H]
$\sim$0.25) (Ness et al., 2013), while the disc has, on average, solar [Z/H]
and a light-weighted age of a few Gyr (Casagrande et al., 2011); for more
details of the properties of the Milky Way, see Venn et al. (2004), Bland-
Hawthorn & Gerhard (2016), Duong et al. (2019) and Rojas-Arriagada et al.
(2019). Are these differences between the Milky Way bulge and disc typical of
other galaxies? What are the mechanisms that produce these differences?

While initial studies of the stellar population of galaxies focussed on early-
type galaxies, it has now been shown that it is possible to extend single
stellar population fitting to galaxies with significant star formation
(Peletier et al., 2007; Ganda et al., 2007; Sánchez-Blázquez et al., 2014).
These studies found that the correlations observed in early-type galaxies,
like the metallicity-mass relation, generally extend to late-type galaxies,
though the scatter in the late-type population is significantly larger than in
early-type galaxies.

Models of galaxy formation involving the monolithic collapse of a gas cloud
usually produce relatively large [Z/H] gradients, decreasing from the centre
of the galaxy (Matteucci & Francois, 1989; Goetz & Koeppen, 1992; Pipino et
al., 2010); Pipino et al. (2010) typically find steep gradients of around
$-$0.3 dex in metallicity per decade variation in radius. In contrast, models
involving multiple mergers produce much flatter gradient profiles, with slopes
a factor of 2–3 smaller (Di Matteo et al., 2009; Taylor & Kobayashi, 2017).

There is an extensive literature supporting the conclusion that the inner
regions of galaxies are more metal rich than the outer regions. Studies
finding negative metallicity gradients in galaxies include Moorthy & Holtzman
(2006), MacArthur et al. (2009), Sánchez-Blázquez et al. (2011), Sánchez-
Blázquez et al. (2014),González Delgado et al. (2015), Goddard et al. (2017),
Zheng et al. (2017), Santucci et al. (2020) and Zibetti et al. (2020). Studies
reporting specifically that bulges are more metal-rich than discs include
Johnston et al. (2012), Johnston et al. (2014), Ganda et al. (2007), Fraser-
McKelvie et al. (2018), Tabor et al. (2019), Barsanti et al. (2021) and
Johnston et al. (2022). Nonetheless, the literature is not unanimous in
finding more metals in the central regions of galaxies, with different results
reported by Goddard et al. (2017), Ferreras et al. (2019), Domínguez Sánchez
et al. (2020) and Pak et al. (2021).

There is a similarly extensive literature showing that the stars in the inner
regions of galaxies tend to be older. Studies finding negative age gradients
in galaxies include Sánchez-Blázquez et al. (2014), González Delgado et al.
(2015), Goddard et al. (2017), Zheng et al. (2017) and Domínguez Sánchez et
al. (2020). Studies reporting specifically that bulges are older than discs
include Fraser-McKelvie et al. (2018), Pak et al. (2021) and Johnston et al.
(2022). However, a substantial number of studies find that the bulge and disc
have similar stellar ages (Tabor et al., 2019; Ferreras et al., 2019; Sánchez-
Blázquez et al., 2011, 2014; Goddard et al., 2017; Zibetti et al., 2020;
Barsanti et al., 2021; Johnston et al., 2022), while others report the inner
regions of galaxies are younger than the outer regions (Johnston et al., 2012,
2014; Goddard et al., 2017; Fraser-McKelvie et al., 2018; Santucci et al.,
2020).

Thus, although most results support bulges being older and more metal-rich
than discs, there are diverging results (the Appendix gives a more detailed
summary of the conclusions reached by the various studies mentioned above). At
least some of the variation in the results reported in the literature is
likely to be due to different observational methods or measurement techniques:
long slit versus integral field spectroscopy; different ways of measuring ages
and metallicities; different ways of measuring gradients; also different
definitions of bulges and discs. Moreover, the differing galaxy sample
properties, such as the distribution of stellar mass, morphology, or
environment, also play a significant role in explaining the differing
conclusions.

The goal of this work is to create a comprehensive study of the stellar
population properties of galaxy bulges and discs using a large sample of
galaxies. To this end, we take galaxy samples from the IFU SDSS-IV MaNGA
survey spanning a wide variety of galaxy types and study the stellar
properties of their bulges and discs. Rather than consider gradients, we
compare instead the stellar ages and metallicities integrated over the bulge
and disc components, determined directly from their Sérsic profiles. This
approach has the significant advantage of always comparing the integrated
stellar populations from consistently-defined morphological components of
galaxies, rather than trying to map ’gradients’ that may simply reflect the
mixture of these components as a function of radius. The disadvantage is that
it is harder to compare to the existing literature, which usually reports
stellar population variations in terms of radial gradients. While the 1D
photometric profile decomposition used here does not split the bulge and disc
light as precisely as a full 2D photometric image bulge-disk decomposition, it
has the advantage (at present) of being applicable to larger, more
representative samples of galaxies from which firm conclusions can be drawn.

This paper has the following structure. In Section 2 the galaxy data is
introduced, starting with a summary of the SDSS-IV MaNGA data (§2.1), then
detailing how the bulge and disc regions were selected (§2.2), describing how
the ages and metallicities were derived from the spectra using stellar
population models (§2.3), the selection cuts of the sample (§2.4) and
providing the other properties of the sample (§2.5). Section 3 compares the
stellar populations of bulges and discs, first across the entire sample (§3.1)
and then for subsamples based on stellar mass, specific star formation rate,
morphology, the Sérsic index of the bulge component, and environment (§3.2).
Section 4 compares the results to the literature and discusses likely physical
causes. Section 5 summarises the conclusions of the paper. The Appendix
contains a more detailed review of the literature studies referenced in the
introduction. We adopt a value of 70 km s-1 Mpc-1 for the Hubble constant.

##  2 Data

###  2.1 MaNGA Data

The Sloan Digital Sky Survey (SDSS) Mapping Nearby Galaxies at APO (MaNGA) is
a large optical integral-field spectroscopy survey of low-redshift galaxies
spanning a broad range in stellar mass, star formation rate, Sérsic index, and
morphology (Bundy et al., 2015; Drory et al., 2015). It is being carried out
using the 2.5m telescope at the Apache Point Observatory (APO; Gunn et al.,
2006) using the BOSS spectrographs (Smee et al., 2013). There are 17 IFUs,
which are deployed simultaneously across the 7 deg2 field of view. The IFUs
range in diameter from 12 arcsec to 32 arcsec. The survey sample has a roughly
flat logarithmic mass distribution of galaxies with stellar masses greater
than 109 M⊙ (Law et al., 2015; Wake et al., 2017). The wavelength coverage is
from 3500 Å to 10000 Å with spectral resolving power $R$$\sim$2000 giving an
instrumental resolution of $\sim$60 km s-1. The spatial resolution is 2.5
arcsec FWHM after combining the dithered images. The galaxies lie within the
redshift range $0.01<z<0.15$ (Yan et al., 2016a). Observations are 3 hours in
length and the signal-to-noise ratio (SNR) is 4–8 Å-1 at 1.5$R_{\rm e}$.
Details on the data reduction pipeline can be found in Law et al. (2016) and
Yan et al. (2016b). For this analysis SDSS Data Release 16 (DR16) was used
(Ahumada et al., 2020). Morphologies were available for most of the sample of
galaxies from Fischer et al. (2019) using deep learning analysis. Elliptical
galaxies were removed from the sample by removing galaxies that had TTYPE$<$0
and S0 probability P_S0$<$0.5. Stellar masses were taken from the mangatarget
table. These were the NSA Elepetro masses that are K-corrected fits from the
elliptical Petrosian fluxes (Blanton et al., 2011). Star formation rates for
the sample were obtained from SED fitting given in the GALEX-SDSS-WISE Legacy
Catalog (GSWLC, Salim et al., 2016). Environmental data for the galaxies was
taken from the GEMA-VAC (Galaxy Environment for MaNGA Value Added Catalog).
This catalog also details whether the galaxies are centrals or satellites. A
catalog with the spectroscopic redshifts for each spaxel of the MaNGA data
cubes (Bolton et al., 2012; Talbot et al., 2018) was also used to correct the
spectra for kinematic effects.

###  2.2 Selecting Bulge and Disc Regions

Figure 1: Definition of ‘bulge’ and ‘disc’ regions. The bulge is defined to be
the bulge-dominated region, where the Sérsic luminosity profile of the bulge
component is brighter than the Sérsic luminosity profile of the disc
component; conversely for the definition of the disc (i.e. the disc-dominated
region). The figure is linear in flux. Although the figure illustrates this in
terms of the 1-dimensional profile, the definitions are actually based on the
2-dimensional profile.

Both one- and two-component Sérsic luminosity profile fits were available for
this sample from the MaNGA PyMorph DR15 photometric catalogue (Fischer et al.,
2019) in g, r and i band. Usually the two-component Sérsic profiles have one
profile with Sérsic index $n>1$ (the bulge) and the other $n=1$ (an
exponential profile, the disc). However, some galaxies that have light
profiles turning sharply downwards at large radii were fit with a $n=1$
profile for the central region and $n<1$ profile for the outer region.

To select the regions defined as bulge and disc, the r band two-component
model profiles were used. First, the point spread function for the observation
was taken from the header parameter RFWHM for the MaNGA galaxies. This is the
reconstructed FWHM in r-band and typically has value $\sim$2.5 arcsec. The
reconstructed FWHM in g-band and i-band usually differs from this value only
in the 2nd decimal place. Images were created based on the Sérsic profiles,
with the bulge and disc Sérsic profile convolved with a Gaussian to take into
account the smoothing due to the point spread function. The bulge-dominated
(disc-dominated) regions are simply defined as those where the Sérsic profile
representing the bulge is brighter (fainter) than the Sérsic profile
representing the disc. Figure 1 provides a visual representation of this as a
function of radius, although the actual decomposition is done in two
dimensions. Near the change-over between bulge- and disc-dominated regions,
there is still significant light from the other component; this contamination
blurs the results of our simple dichotomy somewhat and means that the true
differences between the bulge and disc components are likely stronger than
those measured here. The practical advantage of this method of decomposition
is that it can be done at lower signal-to-noise than other methods and
requires minimal assumptions about the properties of the disc and bulge. The
interpretational advantage is that, whereas gradients conflate the admixture
of the two components’ stellar populations with their relative physical
scales, this method compares the integrated stellar populations of the two
components independently of their physical sizes.

Galaxies were removed from our sample if the bulge Sérsic profile was
everywhere greater than the disc Sérsic profile, or vice versa. Galaxies were
also removed if the disc Sérsic profile was greater than the bulge at the
centre of the galaxy. Where there were repeat observations of the same galaxy,
the data cube with the highest signal-to-noise in the disc spectrum was used.

The selected bulge and disc regions of the MaNGA data cubes are summed to
convert them to one-dimensional spectra, increasing the signal-to-noise. One-
dimensional spectra are also made from the summed spectra of the entire data
cubes for comparison.

###  2.3 Stellar Population Model Fitting

Stellar population ages and metallicities are measured from the summed, one-
dimensional bulge and disc spectra using a full spectral fit based on
theoretical stellar population models from the Medium resolution INT Library
of Empirical Spectra (MILES; Sánchez-Blázquez et al., 2006; Vazdekis et al.,
2010, 2015), BaSTI isochrones (Pietrinferni et al., 2004, 2006), and a
Chabrier (2003) initial mass function. Measurements of the stellar population
ages and metallicities are also made on the combined overall galaxy spectrum.

The process begins with de-redshifting the complete MaNGA spectrum. A first
fit to the spectrum is then made using the penalised pixel-fitting code (pPXF;
Cappellari, 2017) and the nominal variances in the spectrum. Based on the
$\chi^{2}_{\rm reduced}$ of the first fit, the variances are rescaled to give
$\chi^{2}_{\rm reduced}=1$. With this improved noise estimate, a second pPXF
fit is performed to identify any remaining bad pixels (including emission-line
features) using 3 $\sigma$-clipping. The pixels identified as bad or
containing emission lines are then replaced by the best-fit model from this
second fit. The strong sky line at 5577 Å is masked (over 5565–5590 Å) and 13
common emission lines are also masked using the pPXF CLEAN keyword.

After these pre-processing stages, the masked spectrum is fit using a linear
combination of the single-instantaneous-burst MILES synthetic population
templates and a degree-10 Legendre multiplicative polynomial. The role of the
multiplicative polynomial is to correct the shape of the continuum and account
for dust extinction. The MILES templates cover an approximately regular grid
that spans a metallicity range $-2.27\leq$ [Z/H] $\leq 0.40$ and a log age
range of -1.52 Gyr to 1.151 Gyr (unlogged age range of 0.03 Gyr $\leq$ age
$\leq 14.0$ Gyr). The synthetic population templates have solar [$\alpha$/Fe]
abundances. When fitting, each template is assigned a weight and the effective
stellar age and [Z/H] are inferred from the combination of weights; both
light-weighted and mass-weighted fits are performed. Light-weighted
measurements are biased towards the light from bright, massive stars that have
short lifespans; mass-weighted measurements are biased towards the stars that
make up most of the stellar mass, which tend to have lower stellar masses and
are long-lived. For more details on the process of stellar population fitting
see Barone et al. (2020), whose procedures we follow here.

The random errors on the ages and metallicities were generated by taking
template spectrum and adding noise to them. The stellar ages and metallicities
were then measured from these noisy spectra and the variation in the results
computed. A relationship between error and signal-to-noise ratio of the
spectrum was created from this procedure and used to estimate the error for
the ages and metallicities for the galaxy spectra. Based on this work a
minimum continuum signal-to-noise ratio of 15 per pixel was required in the
bulge and disc spectra for the galaxies to be included in the stellar
population sample. At a signal-to-noise ratio of 15 per pixel, the error for
the light-weighted parameters were 0.048 dex in [Z/H] and 0.048 log Gyr in log
age, while for the mass-weighted parameters the error were 0.056 dex in [Z/H]
and 0.055 log Gyr in log age. Besides the random errors, there are systematic
errors that affect the results that cannot be easily estimated.

###  2.4 The Selection Cuts of the Sample

A total of 4857 data cubes were downloaded from DR16. Of these, 4672 have
Sérsic profiles listed in the MaNGA pymorph tables. After removing galaxies
that have undetermined Sérsic values for the bulge ($n=-999$), 4266 data cubes
remain. Galaxies that have Sérsic profiles offset more than 2.5 arcsec from
the centre of the data cubes were removed, leaving 4239 data cubes. Of these
712 had their bulge Sérsic profile always larger than the disc. 531 of these
are elliptical galaxies and the rest of them should probably be labelled as
elliptical galaxies as they have a not very significant disc. These were
removed from the sample leaving 3527. 1137 galaxies had their disc Sérsic
profile always larger than the bulge and 229 had the disc Sérsic profile
larger than the bulge at the centre of the galaxy. These galaxies are probably
best labelled as galaxies without bulges. These 1366 cases were removed from
the sample, leaving 2161 data cubes. For some data cubes the stellar
population analysis failed due to poor signal-to-noise in the spectra, giving
2026 useable data cubes. At this point, multiple observations of the same
galaxy were removed, by adopting the data cube with the highest signal-to-
noise in the disc spectrum, leaving 2007 unique galaxies. A signal-to-noise
cut in both the bulge and disc spectra of 15 per pixel reduced the sample of
galaxies to 1361. Selecting galaxies that had a measurable ESWEIGHT for
completeness corrections left 1342 galaxies. Removing galaxies classified as
ellipticals from the sample reduced this to 1164. This sample was visually
inspected for multiple objects in the MaNGA field of view. There were 106
galaxies with a companion within the field of view, either a star, a nearby
galaxy, or a hot spot in the galaxy. As these could potentially bias the
measurement of the stellar population of the disc, these objects were removed
from the sample, leaving 1058 galaxies. Star formation rates are available for
891 of these galaxies, nearest neighbour measurements for 762, and
central/satellite classifications for 1022.

###  2.5 Other Properties of the Sample

Figure 2: The unweighted histograms of various properties of the galaxies in
the final sample. Top to bottom: (i) log stellar mass; (ii) log specific star
formation rate; (iii) Sérsic indices for the galaxy bulges; and (iv) projected
distance to the 5th nearest neighbour (PD5NN) in Mpc, as a measure of
environmental density.

Figure 2 shows the distributions of various properties for galaxies in the
final sample. The top panel shows that the sample has a peak in log stellar
mass at $\log\,M/M_{\odot}\sim 10.85$, with a long tail to lower masses. The
second panel shows the specific star formation rate (sSFR) distribution has a
shape similar to the log stellar mass distribution, with a peak at $\log({\rm
sSFR/year})\sim-10.75$ and a long tail to lower specific star formation rates.
The third panel gives the distribution of the Sérsic index of the bulge,
showing many galaxies with ‘bulge’ Sérsic index of 1, the fixed index used for
galaxies with light profiles that go sharply downwards at large radii (as
mentioned previously). The bottom panel shows the projected distance to the
5th nearest neighbour (PD5NN) in Mpc, a measure of the local environmental
density; there are few galaxies at the highest densities (PD5NN below 0.2
Mpc), indicative of clusters, but more in lower-density regions.

The red histograms in Figure 3 show the distribution of the fraction of light
in the bulge region (i.e. where the bulge profile dominates) that actually
belongs to the bulge component (as computed from the two-component fit) for g,
r and i band. The blue histogram in the figure shows the distribution of the
fraction of light in the disc region (i.e. where the disc profile dominates)
that actually belongs to the disc component for the same bands. By definition
there is nothing below 50% in the r band sample as these galaxies would have
been classified as bulge or disc dominated and removed from our sample. As the
selection is being done in r band it is possible for the g and i band
measurements to be below 50%. The very low percentages for some of the g and i
band galaxies are when the Sérsic profiles differ substantially between bands,
usually in the orientation of the galaxy.

The level of cross-contamination between components is significant (it should
be noted that this is the amount of light in each region according to the
Sérsic profiles; the actual stellar population of the bulge and disc may not
follow the profiles exactly.) This contamination of bulge by disc and disc by
bulge will tend to blur out any physical differences between bulge and disc
properties, so the actual trends can only be stronger.

There is a wavelength-dependant component to contamination, with the bulge
having higher percentages for i band over r band and with the disc having
higher percentages for g band over i band. This is similar to the differences
in the bulge to total ratio found by Schulz et al. (2003) for U, B, V, I
bands, i.e. increasing bulge to total ratio as one moves to redder bands. This
wavelength contamination shift will effect the stellar population parameters
measured especially as they refer to other properties like Sérsic index,
stellar masses and specific star formation that also vary with wavelength. To
quantify this effect stellar population models were run on bulge and disc
spectra selected using the g and i band Sérsic profiles rather than r band.
This is discussed in Section 3.2 after the main results have been presented.

Figure 3: The number of galaxies with a given percentage of light contributed
by the bulge in the bulge-dominated region is shown by the red histogram in g,
r and i band. The number of galaxies with a given percentage of light
contributed by the disc in the disc-dominated region is shown in the blue
histogram in g, r and i band. The remaining light in each region is the
contaminant from the other component, assuming that the Sérsic profile
accurately traces the light from the bulge and disc. The median contribution
from the bulge in the bulge-dominated region in r band is 59%, while the
median contribution from the disc in the disc-dominated region in r band is
71%

.

##  3 Comparison of bulge and disc stellar populations

###  3.1 Results for the entire sample

The left panel of Figure 4 shows the distribution of the overall stellar [Z/H]
for the sample of galaxies, with the blue histogram showing the light-weighted
[Z/H] and the red histogram the mass-weighted [Z/H]. These overall stellar
[Z/H] values are measured from the spectrum for the whole galaxy before the
decomposition into bulge and disc regions. The light weighted distribution is
offset from the mass weighted distribution and the mass weighted distribution
has more values at higher metallicity than the light weighted distribution.
Both have a sharp decline at higher metallicities with long tails at lower
metallicities.

Figure 4: Left: Histogram of overall stellar [Z/H] for galaxies in the sample.
The blue histogram is the overall light-weighted stellar [Z/H] and the red
histogram is the overall mass-weighted stellar [Z/H]. Right: Histogram of
overall stellar log ages for galaxies in the sample. The blue histogram is the
overall light-weighted stellar log ages and the red histogram is the overall
mass-weighted stellar log ages. In both panels the median errors for the
sample are displayed. This is the random error. There is a larger systematic
error that is not as easily calculated. The median signal to noise for the
sample is 52.

The blue histogram in the right panel of Figure 4 shows the distribution of
the overall light-weighted stellar log age for the sample of galaxies from the
spectra for the whole galaxy before decomposition into bulge and disc samples.
The sample has a fairly uniform distribution until you reach the highest ages
where there is a definite increase. The red histogram in the right panel of
Figure 4 shows the distribution of the overall mass-weighted stellar age for
the sample of galaxies; again, this is from the spectra for the whole galaxy
before decomposition into bulge and disc samples. This distribution is
strongly peaked at high log ages. The differences between the light- and mass-
weighted stellar metallicities are fairly modest but for stellar ages there
are very different results for light-weighted and mass-weighted ages. The era
of most recent star formation, as determined from the high-mass stars measured
by the light-weighted ages, varies across the full range of stellar ages and
indicates when the last major burst of star formation occurred, whereas the
era of peak star formation, as determined from the low-mass stars measured by
the mass-weighted ages, is strongly biased towards very old ages.

Figure 5: Left: Histogram of differences between the stellar [Z/H] of the
bulge and the disc for the galaxies in the sample. The blue histogram shows
the differences in light-weighted [Z/H]; the red histogram shows the
differences in mass-weighted [Z/H]. The arrow is the approximate value of the
difference between bulge [Z/H] and the disc [Z/H] for the Milky Way. The error
bars are the median error bars for the sample. Right: Histogram of the
differences between the stellar log ages of the bulge and the disc for the
galaxies in the sample. The blue histogram shows the differences between the
light-weighted stellar log ages; the red histogram shows the differences
between the mass-weighted stellar log ages. The arrow is the approximate value
of the difference between bulge age and the disc age for the Milky Way
(Casagrande et al., 2011; Ness et al., 2013). The error bars are the median
error bars for the sample. The median signal-to-noise for the bulge sample is
62 and for the disc sample is 32.

The blue histogram in the left panel of Figure 5 shows the distribution of
differences between the bulge and disc light-weighted stellar metallicities
for the sample of galaxies. The differences are generally positive and the
distribution is strongly peaked around [Z/H] $\sim$ 0.2 dex, showing that the
bulge is usually slightly more metal-rich than the disc for this sample. The
red histogram in the left panel of Figure 5 shows the distribution of the
differences between the bulge and disc mass-weighted stellar metallicities for
the sample of galaxies. The distribution is similar to the light-weighted one,
with a peak around [Z/H] $\sim$ 0.2 dex. As shown by the arrow in this figure,
the Milky Way is a fairly typical galaxy in terms of the metallicity
difference between its bulge and its disc as derived from Casagrande et al.
(2011) and Ness et al. (2013).

The blue histogram in the right panel of Figure 5 shows the distribution of
the difference between the bulge and disc light-weighted stellar log ages for
the sample of galaxies. The peak of the distribution is slightly above zero
with well defined wings at greater differences. The red histogram in the right
panel of Figure 5 shows the distribution of the difference between the bulge
and disc mass-weighted stellar log ages for the sample of galaxies. This
distribution also peaks slightly above zero but the distribution is more
peaked and narrower than the light-weighted distribution. As shown by the
arrow in this figure, the Milky Way, with its bulge significantly older than
its disk, is a somewhat atypical galaxy in terms of the age difference between
bulge and disc, lying in the positive tail of the distribution (Casagrande et
al., 2011; Ness et al., 2013). It should be noted here that the value for the
Milky Way is very uncertain and that different methods were used to measure
the ages in the Milky Way and in the galaxy sample (with the Milky Way bulge
and disc both showing a range in stellar ages, $\pm 2$ Gyr). However, the
comparison is interesting, despite its limitations.

###  3.2 Results for subsamples

The number of galaxies in the sample is sufficiently large that it can be
broken down into subsamples that still have reasonable statistical power. We
split the full sample into contrasting pairs of subsamples based on: (i) bulge
Sérsic index, with a split at $n=1.5$; (ii) specific star formation rate, with
a split at $10^{-11}$ yr-1; (iii) stellar mass, with a split at
$M_{\star}=5\times 10^{10}M_{\odot}$; (iv) visual morphology, split into early
types and late types; (v) environment, as determined by the projected distance
to the 5th nearest neighbour, with a split at PD5NN = 0.7 Mpc; and (vi)
whether the galaxy is a central or satellite galaxy. The Sérsic, stellar mass
and projected distance splits were chosen so there would be roughly equal
numbers of galaxies in each category. The sSFR split was chosen to try to
separate the strongly star-forming main sequence from weakly star-
forming/quiescent galaxies, while still having large enough samples for
statistics.

For each subsample, the weighted median values for the stellar population
results for the sample of galaxies were determined, with weights correcting
using the MaNGA esweights (i.e. the medians are for volume-limited samples).
The errors on the weighted medians were estimated using a bootstrap method: a
random sample of values the same size as the observed sample was selected from
the observed values (with replacement) and the weighted median calculated for
this selection; this was repeated 100 times and the standard deviation of
these weighted median values was used as the estimate of the standard error on
the weighted median.

Figure 6: Left: The volume-corrected median values (and estimated
uncertainties) for the difference in stellar metallicity between the bulge and
disc for various subsamples. The blue points are the median values of the
difference in the light-weighted stellar [Z/H]; the red points are the median
values of the difference in the mass-weighted stellar [Z/H]. Right: The
volume-corrected median values (and estimated uncertainties) for the
difference in stellar log age between the bulge and disc for various
subsamples. The blue points are the median values of the difference in the
light-weighted stellar age; the red points are the median values of the
difference in the mass-weighted stellar age. In all cases the number of
galaxies in each sample is written on the right. PD5NN is the projected
distance to the 5th nearest neighbour.

The left panel of Figure 6 shows the volume-corrected median differences in
the stellar metallicities, [Z/H], of the bulge and disc in the various
subsamples of galaxies, for light-weighted (blue) and mass-weighted (red)
estimates. For the whole sample, the bulges are more metal-rich than discs by
around 0.15 dex for both light-weighted and mass-weighted measurements. This
pattern of the bulge having more metals than the disc continues throughout the
subsamples. Subsamples based on Sérsic index mostly agree well with the entire
sample measurements, except for the mass-weighted measurement for galaxies
with low $n$, where the difference is higher than the other measurements (by
$1.9\sigma$ larger). The light-weighted high specific star formation sample
has higher metallicity difference than the low specific star formation sample
(by $2.5\sigma$ larger). The mass-weighted specific star formation values
agree with each other. Both the light-weighted and mass-weighted metallicity
measurements for high stellar mass samples are higher than the low stellar
mass measurements (light-weighted $7.3\sigma$ higher, mass-weighted
$4.2\sigma$ higher). There is no statistical difference between the early- and
late-type subsamples, nor between subsamples based on the projected distance
to the 5th nearest neighbour (PD5NN). There is a difference between the
central galaxy and satellite subsamples, with centrals having more metal-rich
bulges than discs relative to satellites (light-weighted $3.6\sigma$ higher,
mass-weighted $4.3\sigma$ larger). One might suspect this trend is due to
central galaxies tending to be more massive than satellite galaxies, but the
distributions of stellar masses for centrals and satellites in our sample are
similar, so the difference seen here is not simply a mass effect.

The right panel of Figure 6 shows the volume-corrected median differences in
the stellar ages of the bulge and disc in the various subsamples of galaxies,
for light-weighted (blue) and mass-weighted (red) estimates. For all galaxies,
for both light-weighted and mass-weighted measurements, the bulges are
slightly older than the discs with the light-weighted measurement higher than
the mass-weighted measurement (the light-weighted measurement is $3.9\sigma$
above zero and the mass-weighted measurement is $6.5\sigma$ above zero). For
subsamples based on Sérsic index, most agree with each other except the low
Sérsic light-weighted measurement which is larger than the others ($4.1\sigma$
larger). The higher specific star formation sample has larger age difference
than the low specific star formation sample with the light-weighted sample
having the greater difference (light-weighted $3.9\sigma$ larger, mass-
weighted $4.7\sigma$ larger). While the mass-weighted sample shows no
significant difference between the high and low stellar mass samples the light
weighted sample does, with the high stellar mass sample having a larger log
age difference than low stellar mass sample ($5.8\sigma$ larger). Early-type
galaxies tend to have bulges of similar ages to their disc, while late-type
galaxies have bulges significantly older than their discs (light-weighted
$6.6\sigma$ larger, mass-weighted $4.2\sigma$ larger). There is no trend with
projected distance to the 5th nearest neighbour, with both low- and high-
density regions having similar medians. Centrals and sattelites have similar
age differences between their bulges and discs.

It is interesting to return to the values for the Milky Way at this point, as
we can now compare directly to the subsample that best describes our Galaxy.
The Milky Way has a measured star formation rate of $1.65\pm 0.19$ M⊙ yr-1 and
a stellar mass of $(6.08\pm 1.14)\times 10^{10}$ M⊙(Licquia & Newman, 2015).
This gives it a specific star formation rate of $(2.71\pm 0.60)\times
10^{-11}$ yr-1. The Milky Way thus falls in our high-stellar-mass and high-
specific-star-formation-rate subsamples. The [Z/H] difference of around 0.25
dex (Ness et al., 2013; Casagrande et al., 2011) between the bulge and disc of
the Milky Way is slightly higher than, but still close to, the median values
for these samples; this is also true of the late-type and central subsample
measurements. In terms of [Z/H] difference between bulge and disc, the Milky
Way galaxy is close to a representative example. By contrast, the age
difference between the bulge and disc of the Milky Way, at around log age of
0.4 Gyr (Ness et al., 2013; Casagrande et al., 2011), is significantly larger
than the median value for the high sSFR, high stellar mass, late-type or
central subsamples (for either light-weighted or mass-weighted measurements).
Galaxies with bulge–disc age differences as high as the Milky Way are uncommon
in our volume-corrected sample. González Delgado et al. (2015) found that
Milky Way-like galaxies stand out as those with the steepest radial age
profiles. Additionally, the Milky Way has a rather uncommon merging history,
with two fairly large current satellites and no mergers for the past several
Gyr (Evans et al., 2020).

All of these results are affected by the contamination from the overlap of the
bulge and disc stars, which will tend to reduce the apparent stellar
population differences between the ‘bulge-dominated’ and ‘disc-dominated’
regions. It is therefore likely that the underlying physical differences are
in fact larger than those we measure. The Milky Way’s bulge–disc stellar age
difference may not be as significant an outlier if this contamination effect
is taken into account.

The selection here was done using the r band Sérsic profiles. As mentioned
above in Section 2.5 this analysis was repeated using selection with the g and
i band Sérsic profiles. The values for the medians were similar for the g and
i band selections nearly always within $1\sigma$ of the r band medians except
for a few cases where they were within $2\sigma$. There was no clear pattern
in the values moving from g to r to i band. The band used to select the
spectra seems to have minimal effect on the stellar population results.

##  4 Discussion

The result for the full galaxy sample, that bulges have more metals than
discs, agrees well with what is generally found in the literature (Johnston et
al., 2012, 2014; Tabor et al., 2019; Moorthy & Holtzman, 2006; MacArthur et
al., 2009; Sánchez-Blázquez et al., 2011, 2014; González Delgado et al., 2015;
Goddard et al., 2017; Zheng et al., 2017; Zibetti et al., 2020; Barsanti et
al., 2021; Parikh et al., 2021; Johnston et al., 2022). Although most
literature studies measure the gradient of metallicity with respect to radius,
the sign of this gradient can broadly be interpreted in terms of comparing
bulge-dominated and disc-dominated regions. Barone et al. (2018, 2020)
established that for galaxies as a whole the gravitational potential is the
primary regulator of stellar metallicity, through its relationship to the gas
escape velocity. It is a small step to apply this result to the bulge (with
its higher gravitational potential) and disc (with its lower gravitational
potential) to explain the difference seen here in their metallicity.

The higher value in bulge–disc metallicity difference for mass-weighted
measurements of low-Sérsic-index galaxies is an artifact of the completeness
correction. While in general taking the medians without the completeness
correction gives results similar to the ones with the correction, this result
in the exception. Without the completeness correction the bulge–disc
metallicity difference for mass-weighted measurements of low-Sérsic-index
galaxies is similar to the other results. It is unclear why this anomaly
occurs: there is no obvious physical explanation and the result is only
$1.9\sigma$, so it may just be random chance.

The light-weighted high specific star formation sample has a larger
metallicity difference than the low specific star formation sample, though it
is only a $2.5\sigma$ larger. This difference is not present in the mass-
weighted measurements. This may be an artifact of the sample, where our low
sSFR sample tends to have higher stellar mass (i.e. most of the low stellar
mass galaxies in our sample have high sSFR). So this trend may just be a
reflection of the pattern seen for stellar mass.

The higher metallicity difference between bulge and disk for high-stellar-mass
galaxies is likely related to the difference in gravitational potential as
discussed above. The bulges of higher-mass galaxies are better able to hold
onto their metals than lower-mass galaxies that have smaller bulges with lower
gravitational potentials. The light-weighted measurement being higher than the
mass-weighted one indicates that the high-mass stars formed from gas of higher
metallicity that had been retained by the galaxy. These high-mass stars,
having shorter lifetimes, likely formed after the majority of the low-mass
stars that dominate the mass-weighted measurement and so were enriched by
previous generations of stars.

This trend with stellar mass appears to be contrary to the results of
Domínguez Sánchez et al. (2020), who found that galaxies with masses above
$\rm 3\times 10^{10}M_{\odot}$ show little or no metallicity gradients, while
less-massive galaxies in their sample tend to show significant metallicity
gradients. Similarly, Santucci et al. (2020) found negative [Z/H] radial
gradients in their sample, which show evidence of becoming shallower with
increasing stellar mass. The reason for this difference is unclear. The
significant difference between our work and theirs is their use of gradients
versus our bulge-disc measurements. Our bulges are allowed to vary in size,
while the gradients ignore this variation. However, above-mentioned results
based on gradients have agreed with what we found using our method. Our result
does have the natural explanation of increasing gravitational potential with
stellar mass as the source of the differences we see.

We agree with Pak et al. (2021) in finding no trend of bulge–disc metallicity
difference with environment, as measured by projected distance to the 5th
nearest neighbour. However, it should be noted that we are not probing the
very dense environments that are seen in clusters, where there could be such
an effect.

There is a distinction between central and satellite galaxies, with centrals
having higher metallicity difference between bulges and discs. In general
central galaxies tend to be larger than satellite galaxies. However in our
sample this effect is not present, so cannot be the origin of the distinction.
Centrals also tend to have more mergers than satellites and accrete more gas.
This could be an explanation of what we see here with the mergers and the
accretion bringing in fresh gas of lower metallicity that dilutes the
metallicity of stars produced in the disc of centrals. Another difference is
that satellites undergo more quenching than centrals. Star formation could
have quenched in satellite discs before they could produce the same
metallicity difference seen in centrals. It is not possible to distinguish
between these two possibilities with the data we have.

This result differs from that found by Santucci et al. (2020), that
metallicity gradients for fixed stellar mass were similar for satellites and
central galaxies. The most significant difference between our work and
Santucci et al. (2020) is that they limit themselves to passive galaxies only,
while we have star forming galaxies in our sample.

We find in general that the bulges of galaxies are slightly older than their
discs for both light-weighted and mass-weighted measurements. This is
consistent with the measurements made by Sánchez-Blázquez et al. (2014);
González Delgado et al. (2015); Goddard et al. (2017); Zheng et al. (2017);
Fraser-McKelvie et al. (2018); Pak et al. (2021); Johnston et al. (2022). The
age difference is caused by ongoing star formation in the disc after star
formation has ended in the bulge. Star formation is dependent on gas. Gas
could have been used up in the bulge before the disc due to such things like
AGN activity (important in high mass galaxies (Croton et al., 2006)). Gas
could also still be being accreted onto the disc after accretion has ended for
the bulge. Determining the exact scenario would require more and different
data. The actual age difference in linear units is similar between light-
weighted and mass-weighted measurements.

It is not clear why for light-weighted measurements galaxies with lower Sérsic
index galaxies have greater difference between the age of their bulge and
disc. There is no obvious physical reason for this, and one might expect that
galaxies with less defined bulges would have a closer age difference between
bulge and disc. Nonetheless, the result is quite significant, at $4.1\sigma$
result.

It is possible that the age difference seen for specific star formation
samples could be a reflection of the stellar mass difference seen for our
sample. However it could also be due to the fact that galaxies with large sSFR
now are more likely to have the star formation in their discs. This would
decrease the stellar age of the disc relatative to the bulge giving rise to
the age pattern seen here. In particular this would explain the larger
difference seen for the light-weighted measurement that traces the young,
bright stars.

Low-mass galaxies in our light-weighted sample have stellar age differences
between bulge and disc close to zero, while higher mass galaxies have
significant differences between bulge and disc ages. This suggests that
inside-out quenching for high mass galaxies is important, while smaller
galaxies quench across the whole galaxy at the same time. In the more massive
galaxies AGN may be important in stopping the star formation in the bulge
while leaving, for a while at least, star formation continuing in the disc. In
more massive systems, the light-weighted measurement shows a larger difference
in bulge–disc age than the mass-weighted measurement, because the disc is able
to continue forming stars long after the peak of star formation that the mass-
weighted measurement tracks.

Fraser-McKelvie et al. (2018) found that for low-mass lenticular galaxies the
bulge is slightly younger than the disc, while for high-mass lenticular
galaxies the bulge tends to be older than the disc. Pak et al. (2021) and
Johnston et al. (2022) also find similar trends with stellar mass for their
lenticular samples. Domínguez Sánchez et al. (2020) measured stellar
population gradients in MaNGA lenticular galaxies and found the more massive
galaxies showed strong age gradients while the less massive ones showed
relatively flat age gradients. These results match our results for low- and
high-stellar-mass samples, except the galaxies here are a mixture of late and
early types.

Early-type galaxies have measured stellar ages differences between their bulge
and disc close to zero. In early-type galaxies star formation has mostly
stopped, and this result suggest that it stopped in the bulge and disc at the
same time. This result was also found by Goddard et al. (2017); Ferreras et
al. (2019); Tabor et al. (2019); Barsanti et al. (2021) for similar early-type
samples. Domínguez Sánchez et al. (2020) found for lenticular galaxies that
the less massive ones showed relatively flat age gradients as did Johnston et
al. (2022). Our sample has been corrected for stellar mass completeness, so
should be dominated by the low mass galaxies agreeing with these results.
Johnston et al. (2012) and Johnston et al. (2014) actually find bulges younger
than discs for lenticular galaxies going further than our result. This
difference may be due to the fact their sample of galaxies are in clusters
where ram pressure stripping is pushing the disk gas into the centre of a
galaxy where it is used up in a final burst of star formation, making the
bulge stellar population younger than the disc. Our sample does not include
galaxies in such dense environments. A similar result is found by Goddard et
al. (2017) who find a positive age gradient in mass-weighted measurement for
early-type galaxies not limited to clusters. Santucci et al. (2020) also find
age gradients that are slightly positive in early type galaxies. However,
Zheng et al. (2017) found in their mass-weighted measurements that early-type
galaxies have negative age gradients. Pak et al. (2021), in their lenticular
sample, found that most bulges are older than discs. The differences here
between papers are likely due to selection effects on the samples used,
particularly on the definition of what constitutes an early-type galaxy with
some gradient work including ellipticals and lenticular galaxies in the same
sample. In particular Zheng et al. (2017) defines their early type sample
based on Sérsic index which significantly differs from our and other
morphological selection.

Late-type galaxies, in contrast to early-type galaxies, are found to have a
much older bulge than disc in our results. Star formation has continued in the
disc after the bulge has used up all its gas. Goddard et al. (2017) found in
their sample a negative age gradient in light-weighted measurements of late-
type galaxies, supporting our result. However, in their mass-weighted
measurement for late-type galaxies they find a near-zero age gradient. Zheng
et al. (2017) did find a negative age gradients in their mass-weighted
measurements of late-type galaxies in their sample. Sánchez-Blázquez et al.
(2014) find shallow negative stellar age gradient in their late type sample
for light weighted measurements but zero gradient for mass weighted
measurements. González Delgado et al. (2015) found in their sample, negative
age gradients that steepen from E and S0 to Sbc and are shallower from Sbc to
Sd. There is some variation in the results presented here suggesting the
differences in the method or selection of galaxies is playing a role. The fact
that we are comparing to gradients when we are not using them may be a
significant factor. Our mass-weighted measurement is a lot less than our
light-weighed measurment for late type galaxies so could be be seen as near
zero gradient when doing gradient measurements.

There is no trend in bulge-disc age with environment as determined by the
projected distance to the 5th nearest neighbour for the sample, as was found
by Johnston et al. (2022). Pak et al. (2021) found that lenticular galaxies
have larger bulge-disc age differences in higher-density environments. The
difference here can probably be explain with the different determination of
environment with Pak et al. (2021) using projected stellar mass density while
we use projected distance to the 5th nearest neighbour.

Tacchella et al. (2019) in the IllustrisTNG simulation find no significant
metallicity difference between spheroid and disc at any stellar mass and argue
that this must be due to efficient feedback that mixes the metals. This
disagrees with what we find, that there are significant positive metallicity
differences for all stellar masses. Our result seems to imply that the
feedback is not as strong as in the simulations. Tacchella et al. (2019) find,
for mass-weighted measurements of simulated IllustrisTNG galaxies, that
spheroids are older than discs, being log age $\sim$0.022 different at
$\sim$$\rm 11.5\log M_{\odot}$ and log age $\sim$0.15 different at
$\sim$$9\,\rm logM_{\odot}$. For high stellar masses, our mass-weighted
measurement agree with the IllustrisTNG result, but our low stellar masse
measurement does not agree. Here we find that the age difference is similiar
in size to the high stellar mass measurement. This likely indicates that disc
formation occurred generally faster than the simulation predicts in smaller
galaxies. It should be noted that Tacchella et al. (2019) use the kinematics
of a galaxy to select the bulge population which is significantly different
from how we select the bulge population. However they seem confident that
photometric bulge-disc decompositions and kinematical estimates result in
similar conclusions.

##  5 Conclusion

The Mapping Nearby Galaxies at APO (MaNGA) survey is a large, optical,
integral ﬁeld spectroscopic survey of low-redshift galaxies spanning a broad
range in stellar mass, star formation rate, Sérsic index, and morphology. From
this survey, we measure and compare the stellar populations of the bulge- and
disc-dominated regions identified from their Sérsic profiles. Weighted medians
of the metallicity and stellar age difference between bulge- and disc-
dominated regions are measured, taking into account the completeness of the
sample.

For the entire sample, we find that generally bulge regions have higher
metallicities than disc regions in both light-weighted and mass-weighted
measurements. One likely cause of this difference is the greater gravitational
potential within bulges compare to disc that enable them to retain more of
their metals that are produced in star formation. Galaxies with higher stellar
masses tend to have higher metallicity differences between bulge regions and
disc regions than lower stellar mass galaxies, with light-weighted
measurements showing the greatest difference. Again the great gravitational
potential of bulges in galaxies with high stellar masses is the likely reason
for this difference. Early and late-type galaxies have similar metallicity
differences between bulge regions and disc regions. There is no trend in
metallicity difference between bulge regions and disc regions with
environment, at least over the range that the MaNGA sample covers. Central
galaxies tend to have greater metallicity difference between their bulge
regions and disc regions than satellite galaxies at similar stellar mass.
Influx of lower metallicity gas into the disc of centrals either by accretion
or mergers could be the cause. Quenching of satellite discs is the other
possible explanation.

For the entire sample, we find that bulge regions generally have slightly
higher stellar ages than disc regions in both light-weighted and mass-weighted
measurements. Ongoing star formation in the disc fuelled by more gas, possibly
fed by accretion, is the likely cause. AGN activity in the bulge, halting star
formation there, may also be a factor. Galaxies with higher specific star
formation also have greater age differences between bulge and disc. Our lower
stellar mass sample shows little difference between bulge regions and disc
regions stellar ages, while our higher stellar mass light-weighted sample has
a median stellar age difference between bulge regions and disc regions that is
positive. This suggests inside-out quenching for high-mass galaxies, while
smaller galaxies quench across the whole galaxy simultaneously. Early-type
galaxies tend to have bulge regions about the same stellar age as their disc
regions, while late-type galaxies have a significantly positive difference
between bulge regions and disc regions stellar age. We find no trend in
stellar age difference between bulge regions and disc regions with
environmental density for the range that MaNGA probes. Central and satellite
galaxies have similar stellar age differences between their bulge and disk
regions.

The Milky Way has a metallicity difference between bulge and disc that lies
close to the typical results for our sample, indicating that it is a normal
galaxy in this respect. However, the Milky Way has a stellar age difference
between bulge and disc that is significantly higher than the typical result we
find here. This larger age difference may in part be due to the Milky Way
measurement having better separation of the bulge and disc light than is
possible in our measurements.

## Acknowledgements

Funding for the Sloan Digital Sky Survey IV has been provided by the Alfred P.
Sloan Foundation, the U.S. Department of Energy Office of Science, and the
Participating Institutions. SDSS acknowledges support and resources from the
Center for High-Performance Computing at the University of Utah. The SDSS web
site is www.sdss.org.

SDSS is managed by the Astrophysical Research Consortium for the Participating
Institutions of the SDSS Collaboration including the Brazilian Participation
Group, the Carnegie Institution for Science, Carnegie Mellon University, the
Chilean Participation Group, the French Participation Group, Harvard-
Smithsonian Center for Astrophysics, Instituto de Astrofísica de Canarias, The
Johns Hopkins University, Kavli Institute for the Physics and Mathematics of
the Universe (IPMU) / University of Tokyo, the Korean Participation Group,
Lawrence Berkeley National Laboratory, Leibniz Institut für Astrophysik
Potsdam (AIP), Max-Planck-Institut für Astronomie (MPIA Heidelberg), Max-
Planck-Institut für Astrophysik (MPA Garching), Max-Planck-Institut für
Extraterrestrische Physik (MPE), National Astronomical Observatories of China,
New Mexico State University, New York University, University of Notre Dame,
Observatório Nacional / MCTI, The Ohio State University, Pennsylvania State
University, Shanghai Astronomical Observatory, United Kingdom Participation
Group, Universidad Nacional Autónoma de México, University of Arizona,
University of Colorado Boulder, University of Oxford, University of
Portsmouth, University of Utah, University of Virginia, University of
Washington, University of Wisconsin, Vanderbilt University, and Yale
University.

Nicholas Scott acknowledges support of an Australian Research Council
Discovery Early Career Research Award (project number DE190100375). Tania M.
Barone is supported by an Australian Government Research Training Program
Scholarship. Francesco D’Eugenio acknowledges funding through the H2020 ERC
Consolidator Grant 683184 and the ERC Advanced grant 695671 “QUENCH” and
support by the Science and Technology Facilities Council (STFC).

We wish to thank Gary Da Costa, Ken Freeman and Mike Bessell for their
valuable insight. We wish to thank Christopher Delfs for his valuable
assistance. We wish to thank Anne-Marie Weijmans, Mariangela Bernardi,
Benjamin Alan Weaver Rodrigo Tobar Carrizo and Mina Pak for their valuable
help.

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## Appendix

This Appendix gives a detailed summary of the conclusions reached by the
papers listed in the Introduction.

There is extensive literature showing that the inner regions of galaxies have
more metals than outer regions.

  * •

Moorthy & Holtzman (2006) observed 38 galaxies with well-defined bulges and
discs using a long-slit spectrograph. They found that most bulges have
decreasing [Z/H] with increasing radius, and that galaxies with larger central
metallicities have steeper gradients.

  * •

MacArthur et al. (2009) examined eight star-forming, spiral galaxies using
long-slit observations and found a small decrease with radius in [Z/H] for at
least four of their sample.

  * •

Sánchez-Blázquez et al. (2011) found, in long-slit observations of 4 galaxies
ranging from lenticulars to late-type spirals, a decrease in [Z/H] in three of
the four galaxies.

  * •

Johnston et al. (2012) used long-slit observations of nine lenticular galaxies
(S0s) in the Fornax cluster to reveal that, where the bulge and disc could be
distinguished, the bulges have systematically higher metallicities.

  * •

In Johnston et al. (2014), a further 21 lenticular galaxies in the Virgo
cluster were examined using a long-slit spectrograph, and the bulges were
found to be more metal-rich than their surrounding discs.

  * •

Ganda et al. (2007) observed 18 late-type spiral galaxies with an integral-
field spectrograph and showed that [Z/H] decreases from the bulge to the disc.

  * •

IFU observations by Sánchez-Blázquez et al. (2014) of 62 spiral galaxies found
mean metallicity gradients are shallow and negative.

  * •

González Delgado et al. (2015) observed 300 galaxies with integral field
spectroscopy for a wide range of Hubble types as part of the CALIFA survey. In
their sample [Z/H] gradients are mildly decreasing profiles for most Hubble
types except Sd galaxies, which show little decrease; most of their galaxies
have stellar masses larger than 1010M⊙.

  * •

Goddard et al. (2017) reported on observations of 721 MaNGA galaxies showing
that, in light-weighted [Z/H] measurements, early-type galaxies have a
negative gradient.

  * •

Zheng et al. (2017) used MaNGA observations of 1005 galaxies to examine
galaxies of all types from the survey. They find, in mass-weighted
measurements, that the mean [Z/H] gradients are close to zero but slightly
negative for both early and late-type galaxies.

  * •

Fraser-McKelvie et al. (2018) studied 279 lenticular galaxies from the MaNGA
survey and found that the higher mass lenticulars have bulges more metal-rich
than their discs, though in general bulges and discs had similar
metallicities.

  * •

Tabor et al. (2019) studied 273 early-type galaxies from the MaNGA survey and
found that bulges have higher metallicites than discs, which tend to span a
wider [Z/H] range indicating a more varied star formation history compared to
bulges.

  * •

In a sample of 96 passive central galaxies with integral-field spectroscopy
from the SAMI Galaxy Survey measured out to $\sim$2Re, Santucci et al. (2020)
found negative [Z/H] radial gradients and evidence that gradients become
shallower with increasing stellar mass. They also found that metallicity
gradients for fixed stellar mass were similar for satellites and central
galaxies.

  * •

Barsanti et al. (2021) studied 192 lenticular galaxies in the cluster sample
of the SAMI Galaxy Survey using 2D photometric bulge-disk decomposition and
found that most galaxies have more metals in the bulge than disc.

  * •

Zibetti et al. (2020) looked at 69 early-type galaxies from the CALIFA
integral field spectroscopic survey. They found that early-type galaxies
universally had strong, negative metallicity gradients within 1 Re that
flatten out at larger radii.

  * •

Johnston et al. (2022) found in 78 lenticular galaxies in the MaNGA sample
that bulges are generally more metal rich than discs.

However, the literature is not unanimous in finding more metals in the central
regions of galaxies.

  * •

Goddard et al. (2017) found that both light-weighted and mass-weighted
measurements of late-type galaxies showed near zero [Z/H] gradient, and mass-
weighted measurements of [Z/H] gradients in early-type galaxies were also near
zero.

  * •

Ferreras et al. (2019) examined 522 early-type galaxies from the SAMI Galaxy
Survey, looking at the relationships between [Z/H], [Mg/Fe], [C/Fe] and age
gradients compared with velocity dispersion, stellar mass, dynamical mass,
surface stellar mass density, stellar potential and virial test
($\sigma^{2}$/R). They found a near zero [Z/H] gradient in their sample of
galaxies.

  * •

Domínguez Sánchez et al. (2020) measured stellar population gradients in MaNGA
lenticular galaxies by stacking in bins of luminosity and central velocity
dispersion. Galaxies with masses above $\rm 3\times 10^{10}M_{\odot}$ show
little or no metallicity gradients, while less massive ones show a significant
metallicity gradient.

  * •

Pak et al. (2021) used photometric bulge-disk decomposition to separate bulges
and discs for 34 lenticular galaxies from the CALIFA survey. In their sample
they find no no systematic difference in metalllicity between bulges and their
associated discs. They also found that the bulge-disc metallicity difference
does not vary as a function of local density.

The literature includes many studies showing that the stars in the inner
regions of galaxies tend to be older.

  * •

Sánchez-Blázquez et al. (2014) showed that the mean stellar age gradient has a
shallow but negative slope for their sample of spiral galaxies.

  * •

González Delgado et al. (2015) found in their sample, negative age gradients
steepen from E and S0 to Sbc and are shallower from Sbc to Sd. Milky Way-like
(Sbc) galaxies thus stand out as those with the steepest radial age profiles.

  * •

Goddard et al. (2017) found a negative age gradient in lighted-weighted
measurements of late-type galaxies.

  * •

Zheng et al. (2017) found that mass-weighted measurements yield negative age
gradients for both early-type and late-type galaxies.

  * •

Fraser-McKelvie et al. (2018) found that, for massive lenticular galaxies, the
bulge was older than the disc, although for most of their sample they found
that bulge and disc ages were similar.

  * •

In the lenticular sample of Domínguez Sánchez et al. (2020), the more massive
galaxies showed strong age gradients, while the less massive ones showed
relatively flat age gradients.

  * •

In their CALIFA galaxy sample of lenticular galaxies, Pak et al. (2021) found
that most bulges are older than their associated discs. They also find that
for lenticular galaxies the difference between their bulge and disc age
increase with increasing stellar mass. Additionally they find that lenticular
galaxies have larger age differences in higher-density environments.

  * •

Johnston et al. (2022) found in lenticular galaxies that bulges are generally
older than discs. They also found no clear difference in the formation or
quenching processes comparing bulges and discs as a function of galaxy
environment.

Some sources in the literature have found that the bulge and disc have similar
stellar ages.

  * •

Sánchez-Blázquez et al. (2011) found a fairly flat age gradient in their four
galaxy sample.

  * •

The mass-weighted measurements of Sánchez-Blázquez et al. (2014) for late-type
galaxies show zero stellar age gradient.

  * •

Goddard et al. (2017) found a near zero age gradient in the light-weighted
measurements of early-type galaxies and mass-weighted late-type galaxies in
their sample.

  * •

Ferreras et al. (2019) also found near zero age gradient in early-type
galaxies from the SAMI Galaxy Survey.

  * •

One example is the measurements made by Tabor et al. (2019) of early-type
galaxies from the MaNGA survey.

  * •

Zibetti et al. (2020) in the CALIFA survey found age profiles that have
similar ages at the centre and the outskirts of the galaxies, but are U-shaped
in between, with a minimum around 0.4 Re and asymptoting to a maximum age
beyond 1.5 Re.

  * •

Barsanti et al. (2021) found in their cluster, lenticular sample that most
galaxies have bulges and disks with statistically indistinguishable ages,
though there are fair numbers of galaxies with bulge ages both older and
younger than their associated discs.

  * •

Johnston et al. (2022) found in low mass lenticular galaxies ($\rm\leq
10^{10}M_{\odot}$) that bulges and discs tend to have similar ages.

A few sources in the literature show inner region younger than outer regions.

  * •

The Johnston et al. (2012) lenticular galaxy sample showed younger stellar
populations in their bulges than the discs, as did the results of Johnston et
al. (2014). Both of these samples were in clusters, and they suggested that
ram pressure stripping is pushing the disc gas into the centre of the galaxies
where it is used up in a final burst of star formation, making the bulge
stellar population on average younger than the disc.

  * •

Goddard et al. (2017) found in their sample of early-type galaxies that the
mass-weighted stellar age showed an increasing gradient.

  * •

Fraser-McKelvie et al. (2018) found that lower-mass lenticulars tend to have
bulges marginally younger than their discs.

  * •

In a passive central galaxy sample from the SAMI Galaxy Survey, Santucci et
al. (2020) find age gradients that are slightly positive and no difference in
the trend between central and satellite galaxies.


A note on geodesic embeddings of hyperbolic surfaces

into hyperbolic 3-manifolds

Bruno P. Zimmermann

Abstract. We consider the problem of when a closed orientable hyperbolic
surface admits a totally geodesic embedding into a closed orientable
hyperbolic 3-manifold; given a finite isometric group action on the surface,
we consider also an equivariant version of such an embedding. We prove that an
equivariant embedding exists for all finite irreducible group actions on
surfaces; such surfaces are known also as quasiplatonic surfaces; in
particuler, all quasiplatonic surfaces embed geodesically. Finally, we discuss
also the case of reducible group actions.

e-mail<EMAIL_ADDRESS>

1\. Introduction

In the present note, all surfaces and 3-manifolds will be compact, connected
and orientable, all group-actions orientation-preserving. Let $\cal S$ be a
closed hyperbolic surface and $G$ a finite group of isometries of $\cal S$; we
say that the pair $({\cal S},G)$ embeds geodesically if there exists a totally
geodesic embedding of $\cal S$ into a closed hyperbolic 3-manifold ${\cal M}$
such that the action of $G$ on ${\cal S}$ extends to an isometric action of
$G$ on ${\cal M}$; if $G$ is the trivial group, we just say that the
hyperbolic surface ${\cal S}$ embeds geodesically. Moreover, the pair $({\cal
S},G)$ bounds geometrically if ${\cal S}$ is the unique, totally geodesic
boundary component of a compact hyperbolic 3-manifold ${\cal M}$ and the
action of $G$ on ${\cal S}$ extends to an isometric action of $G$ on ${\cal
M}$.

It is shown in [GZ] (see also the survey [Z1]) that various infinite series of
Hurwitz-actions of maximal possible order $84(g-1)$ on a surface of genus $g$
bound geometrically; in particular, the underlying Hurwitz surfaces bound
geometrically. Finite group actions on surfaces of small genus are considered
in [WZ] and [Z2]; for example, the smallest Hurwitz action of order 168 on
Klein’s quartic of genus 3 does not bound any compact 3-manifold with a unique
boundary component but it remains open whether Klein’s quartic bounds
geometrically; however, as a consequence of [Z3, Proposition 1], every Hurwitz
action embeds geodesically.

Lifting a finite group $G$ of isometries of a hyperbolic surface ${\cal S}$ to
its universal covering ${H}^{2}$ we get a Fuchsian group $F$ and an exact
sequence $1\to\pi_{1}({\cal S})\to F\to G\to 1$ (where $\pi_{1}({\cal S})$ is
interpreted as the group of covering tranformations). We are interested here
in the cases where $F$ is a triangle group $(p,q,r)$ (i.e., the orientation-
preserving subgroup of index 2 of the group generated by the reflections in
the sides of a hyperbolic triangle with angles $\pi/p,\pi/q$ and $\pi/r$),
with a presentation

$<x,y\mid x^{p}=y^{q}=(xy)^{r}=1>.$

These are exactly the irreducible actions of finite groups on surfaces (cf.
[Z1]), and we say that the action of $G$ is of triangle type $(p,q,r)$. Since
any two triangle groups of the same type are conugate by an isometry, an
irreducible action of a finite group $G$ determines the hyperbolic structure
of the underlying surface ${\cal S}$ (this is true only for Fuchsian triangle
groups since the Teichmüller or deformation space of any other Fuchsian group
has positive dimension).

We call a hyperbolic surface $\cal S$ semiplatonic if its universal covering
group is a subgroup of finite index in a triangle group $(p,q,r)$; if the
universal covering is a normal subgroup of a triangle group $(p,q,r)$ then the
surface $\cal S$ admits an irreducible action of the factor group
$G=(p,q,r)/\pi_{1}(\cal S)$, and such surfaces are known also as quasiplatonic
surfaces (cf. [SW]).

Theorem 1. Every finite irreducible group action $({\cal S},G)$ of a finite
group $G$ on a closed hyperbolic surface ${\cal S}$ embeds geodesically into a
closed hyperbolic 3-manifold; in particular, every quasiplatonic surface
${\cal S}$ embeds geodesically.

Corollary 1. Every Hurwitz action of maximal possible order $84(g-1)$ on a
closed hyperbolic surface of genus $g$ embeds geodesically into a closed
hyperbolic 3-manifold; in particular, all Hurwitz surfaces embed geodesically.

However, as noted before, not all Hurwitz actions bound geometrically (not
even topologically, see [Z1]).

For the case of semiplatonic surfaces, for the moment we have to pass from
hyperbolic 3-manifolds to hyperbolic 3-orbifolds.

Corollary 2. Every semiplatonic hyperbolic surface embeds geodesically into a
closed hyperbolic 3-orbifold.

It remains open here if we can replace 3-orbifold by 3-manifold but we believe
that this is the case (cf. [R]).

In general, finite reducible actions on surfaces do not embed geodesically
since for each topological type of such an action there is an uncountable
continuum of geometric realizations $({\cal S},G)$ but only countably many can
embed geometrically. However we state the following:

Conjecture. Let $G$ be a finite group of homeomophisms of a closed surface
$\cal S$ of genus $g>1$. Then some geometric realization of $({\cal S},G)$ by
a hyperbolic surface $\cal S$ and a group of isometries $G$ of $\cal S$ embeds
geodesically into a closed hyperbolic 3-manifold.

For irreduciblle actions, there is a unique geometric realization $({\cal
S},G)$ which, by Theorem 1, embeds geodesically. In the next theorem we
consider the case of reducible actions which realize the largest possible
orders after irreducible actions. Let $G$ be a finite group of homeomorphisms
of a closed surface of genus $g>1$. By choosing a hyperbolic structure of the
quotient 2-orbifold ${\cal S}/G$ and lifting to $\cal S$, one realizes $\cal
S$ as a hyperbolic surface on which $G$ acts by isometries. By lifing $G$ to
the universal covering ${H}^{2}$ one obtains a Fuchsian group, and we consider
the case of a quadrangle group $(p,q,r,s)$ which has a presentation

$<x,y,z,w\mid x^{p}=y^{q}=z^{r}=w^{s}=xyzw=1>;$

we say that the action of $G$ is of quadrangle type $(p,q,r,s)$.

Theorem 2. Let $G$ be a finite group of quadrangle type of homeomorphisms of a
surface $\cal S$ of genus $g>1$. Then, for some geometric realization of
$({\cal S},G)$ by a hyperbolic surface $\cal S$ and an isometric action of
$G$, the pair $({\cal S},G)$ embeds geodesically into a closed hyperbolic
3-manifold.

2\. Proof of Theorem 1

i) We give the proof first for the case of an irreducible action of triangle
type $(2,q,r)$, and then describe the changes necessary for the general case
$(p,q,r)$.

Let $({\cal S},G)$ be a finite irreducible group action of triangle type
$(2,q,r)$. Let ${\cal T}(2,q,r)$ be a truncated hyperbolic tetrahedron with
dihedral angles $\pi/2,\pi/q$ and $\pi/r$ (or with labels 2, $q$ and $r$) at
the three pairs of opposite edges, truncated by orthogonal planes at the four
virtual vertices (so ${\cal T}(2,q,r)$ has four hexagonal faces, and four
triangular faces on the truncating planes). Let $T(2,q,r)$ denote the
tetrahedral group associated to ${\cal T}(2,q,r)$, i.e. the orientation-
preserving subgroup of index 2 in the Coxeter group generated by the
reflections in the four hexagonal faces of the tetrahedron. Let $a,b,x,y$
denote the rotations of orders $q,2,2,q$ around the edges with these labels
(suitably oriented). We represent the six labelled edges of the tetrahedron by
a square and its two diagonals, avoiding the intersection point of the
diagonals by an over/under-crossing and removing small open neighborhoods of
the four vertices, with label $r$ at the two upper and lower edges of the
square, label $q$ at the right and the left edge, and label 2 at the two
diagonals; then, using standard methods for the computation of an orbifold
fundamental group, one obtains the following presentation of the tetrahedral
group $T(2,q,r)$ (see [Z4] or [GZ] for similar computations; see also [B]):

$<a,b,x,y\mid
b^{q}=a^{2}=x^{2}=y^{q}=1,\;ab=xy,\;(ab)^{r}=(xy)^{r}=(x^{-1}b)^{r}=1>$

(with the rotations $b$ and $y$ suitably oriented; also,
$((x^{-1}a^{-1}x)y)^{r}=1$). The sugroup of $T(2,q,r)$ generated by $x$ and
$y$ is a triangle group of type $(2,q,r)$, with the presentation given in
section 1 (with $p=2$), and the surjection $\phi:(2,q,r)\to G$ associated to
the action of $G$ on ${\cal S}$ extends to a surjection $\psi:T(2,q,r)\to G$
by setting $\psi(x)=\phi(x)$, $\psi(y)=\phi(y)$, $\psi(a)=\phi(x)$ and
$\psi(b)=\phi(y)$; note that
$\psi((x^{-1}b)^{r})=(\psi(x^{-1})\psi(b))^{r}=(\psi(x)\psi(b))^{r}=(\phi(x)\phi(y))^{r}=\phi(xy)^{r}=1$,
so this is exactly the point where we use that $x$ has order 2.

Let $K\subset T(2,q,r)$ denote the kernel of $\psi$. Let $\tilde{H}^{3}$
denote the hyperbolic 3-space truncated by all images under $T(2,q,r)$ of the
four truncating planes of the tetrahedron ${\cal T}(2,q,r)$. The quotient
$M=\tilde{H}^{3}/K$ is a compact hyperbolic 3-manifold $\cal M$ with an
isometric action of $G$ (the projection of $T(2,q,r)$), and with totally
geodesic boundary consisting of four copies of the hyperbolic surface ${\cal
S}$, each with an action of type $(2,q,r)$ of $G$ (all isometrically
equivalent). To obtain a closed hyperbolic 3-manifold with a geodesic
embedding of $({\cal S},G)$, one can either take the double of ${\cal M}$ and
its $G$-action along the boundary of $\cal M$, or alternatively glue in pairs
the four boundary surfaces of ${\cal M}$ by $G$-equivariant isometries.

This completes the proof of the Theorem 1 for the case of an action of type
$(2,q,r)$. The general case $(p,q,r)$ is similar but we replace the hyperbolic
3-orbifold $\tilde{H}^{3}/K$ by a more complicated one and indicate the
necessary changes in the following (see also the Remark in section 3 for
another proof).

The underlying topological space of the hyperbolic 3-orbifold ${\cal
O}=\tilde{H}^{3}/K$ (compact with four totally geodesic boundary components)
is the 3-sphere minus the interiors of four disjoint 3-balls, its singular set
consists of the three pairs of opposite edges with labels $2,q$ and $r$ of the
truncated tetrahedron ${\cal T}(2,q,r)$. Representing the singular set by a
square with diagonals as before, we associate label $p$ now to the two
diagonals (instead of label 2 as before). This defines a compact hyperbolic
3-orbifold ${\cal O^{\prime}}$ with four totally geodesic boundary components,
each a 2-sphere with three singular points of orders $p$, $q$ and $r$. The two
diagonals have one over/under crossing or half-twist which we replace by $u$
half-twists (for an integer $u$); this defines a 3-orbifold ${\cal O}(u)$
which is obtained by orbifold Dehn surgery on the orbifold ${\cal O^{\prime}}$
(see [Z4] or [GZ] for some figures). By the orbifold version of Thurston’s
hyperbolic Dehn surgery theorem, for large enough values of $u$ the orbifold
${\cal O}(u)$ is hyperbolic again, compact and with totally geodesic boundary
(in fact, we believe that $u=2$ is sufficient, applying the orbifold version
of Thurston’s hyperbolization theorem).

As before, the orbifold fundamental group $\pi_{1}({\cal O}(u))$ is generated
by elements $a,b,x,y$ (see [Z4], [GZ]). We suppose now that $u$ is even; then,
as in the first part of the proof, the surjection $\phi:(p,q,r)\to G$
associated to the action of $G$ on ${\cal S}$ extends to a sujection
$\psi:\pi_{1}({\cal O}(u))\to G$ (which does not work if $u$ is odd unless
$p=2$). Now the proof is completed exactly as in the first case.

Proof of Corollary 2. For a closed hyperbolic surface ${\cal S}$, suppose that
its universal covering group $\pi_{1}({\cal S})$ is a subgroup of finite index
in a triangle group $(p,q,r)$. By taking the intersection of $\pi_{1}({\cal
S})$ with all of its conjugates in $(p,q,r)$, one obtains a normal subgroup of
finite index of $(p,q,r)$ which is the fundamental group $\pi_{1}(\tilde{\cal
S})$ of a finite covering $\tilde{\cal S}$ of ${\cal S}$. Then $\tilde{\cal
S}$ is a quasiplatonic surface of type $(p,q,r)$, with an isometric action of
$G=(p,q,r)/\pi_{1}(\tilde{\cal S})$ associated to a surjection
$\phi:(p,q,r)\to G$ with kernel $\pi_{1}(\tilde{\cal S})$. Let $G_{0}$ denote
the image $\phi(\pi_{1}({\cal S}))$ of $\pi_{1}({\cal S})\subset(p,q,r)$ in
$G$; $G_{0}$ is the covering group of the covering $\tilde{\cal S}$ of ${\cal
S}$, with $\tilde{\cal S}/G_{0}={\cal S}$.

By Theorem 1, $(\tilde{\cal S},G)$ embeds geodesically into a closed
hyperbolic 3-manifold $\cal M$ with an isometric $G$-action. Then ${\cal
M}/G_{0}$ is a closed hyperbolic 3-orbifold with a geodesic embedding of the
surface $\tilde{\cal S}/G_{0}={\cal S}$ (note that $G_{0}$ has no fixed points
in $\tilde{\cal S}$ but might have fixed points in $\cal M$).

2\. Proof of Theorem 2

As before, the action of $G$ on $\cal S$ is defined by a surjection
$\phi:(p,q,r,s)\to G$ with kernel $\pi_{1}(\cal S)$, and we have the
presentation $<x,y,z,w\mid x^{p}=y^{q}=z^{r}=w^{s}=xyzw=1>$ of the quadrangle
group $(p,q,r,s)$.

We consider a hyperbolic 3-orbifold ${\cal O}$ with totally geodesic boundary
whose underlying topological space is $S^{3}$ minus the interiors of 6 small
disjoint 3-balls around the vertices of a standardly embedded octahedron, and
whose singular set $\Sigma$ is the 1-skeleton of the octahedron, truncated at
its six vertices by the six 3-balls. We assign labels $p,q,r$ and $s$ to the
12 edges of the octahedron in the following way.

We consider a planar projection of the 1-skeleton of the octahedron,
consisting of a square with its diagonals representing 8 edges of the
octahdron, a vertex outside of the square connected by four further edges to
the vertices of the square, and also a direction of undercrossing for each of
the 12 (disjoint) edges. It is easy to verify then that one can assign labels
$p,q,r$ and $s$ to the twelf edges such that at four of the vertices of the
projected 1-skeleton of the octahedron, the labels are in cyclic order $p,q,r$
and $s$ (following the chosen directions of the undercrossings), and at each
of the remaining two vertices the four edges have the same label, two times in
the chosen and two times in the opposite direction.

By Andreev’s theorem on hyperbolic polyhedra ([A], [T, chapter 13], [V,
p.111]), the truncated octahedron can be realized in ${H}^{3}$ as a
hyperbolic, orthogonally truncated octahedron, with dihedral angles
$\pi/p,\pi/q,\pi/r$ and $\pi/s$ according to the labels of the 12 edges (since
there are no essential Euclidean 2-suborbifolds). Let $O$ denote the
orientation-preserving subgroup of index 2 in the group generated by the
reflections in the faces of this hyperbolic octahedron. Then $O$ acts on
$\bar{H}^{3}$, obtained by truncating ${H}^{3}$ by the truncating planes of
the octahedron and its images under $O$, and the quotient-orbifold
$\bar{H}^{3}/O$ realizes the hyperbolic 3-orbifold $\cal O$, with orbifold
fundamental group $\pi_{1}({\cal O})\cong O$.

A presentation of the orbifold fundamental group $\pi_{1}(\cal O)$ of $\cal O$
can be obtained in a standard way from the planar projection of the singular
set $\Sigma$ (similar as the Wirtinger presentation of the fundamental group
of the complement of a link in $S^{3}$ is obtained from a planar projection).
The generators correspond to the edges of $\Sigma$, indicated by oriented
small undercrossings of the 12 edges of the planar projection of $\Sigma$.

We define a surjection $\psi:\pi_{1}({\cal O})\to G$ as follows. If an edge
has label $p$, the image under $\psi$ of the corresponding generator is
defined as $\phi(x)$, and for labels $q,r$ and $s$ the images are respectively
$\phi(y),\phi(z)$ and $\phi(w)$.

The orbifold covering of $\cal O$ corresponding to the kernel of $\psi$ is a
compact hyperbolic 3-manifold $\cal M$ with totally geodesic boundary and an
isometric action of $G$. On four of the boundary components of $\cal M$ the
action of $G$ is equivalent to the given action on $\cal S$ (on all other
boundary components there are actions of cyclic subgroups of $G$, of orders
$p,q,r$ or $s$). By doubling $\cal M$ and the action of $G$ along the boundary
we obtain a closed hyperbolic 3-manifold with an isometric action of $G$ and a
geodesic embdding of $({\cal S},G)$, completing the proof of Theorem 2.

Remark. A proof of Theorem 1 can be given along similar lines, avoiding the
surgery argument in the second part of the proof. Let $({\cal S},G)$ be a
finite irreducible group action of triangle type $(p,q,r)$, defined by a
surjection $\phi:(p,q,r)\to G$. If one of the integers $p,q$ or $r$ is equal
to 2, we refer to the proof given in section 2 using a truncated tetrahedron.

If $p,q$ and $r$ are all different from 2, we use a truncated cube instead. We
consider a planar projction of the 1-skeleton of the cube which consists of
two concentrical squares (one contained in the other, with parallel edges)
whose vertices are joined by four further edges (parts of the diagonals of the
larger square). Then, similar as in the proof of Theorem 2, one can label the
12 edges of the cube by $p,q$ and $r$ and choose directions of undercrossings
such that, around each vertex, the labels of the edges are in cyclic order
$p,q,r$. Since we excluded 2 as a label, there are no essential Euclidean
2-suborbifolds (of type (2,2,2,2)). By Andreev’s theorem there is a hyperbolic
relalization in ${H}^{3}$ of the orthogonally truncated cube, and the proof is
then similar to the proof of Theorem 2.

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