# Expanders and growth of normal subsets in finite simple groups of Lie type

Saveliy V. Skresanov The research was carried out within the framework of the
Sobolev Institute of Mathematics state contract (project FWNF-2022-0002).

###### Abstract

We show that some classical results on expander graphs imply growth results on
normal subsets in finite simple groups. As one application, it is shown that
given a nontrivial normal subset $A$ of a finite simple group $G$ of Lie type
of bounded rank, we either have $G\setminus\\{1\\}\subseteq A^{2}$ or
$|A^{2}|\geq|A|^{1+\epsilon}$, for $\epsilon>0$. This improves a result of
Gill, Pyber, Short and Szabó, and partially resolves a question of Pyber from
the Kourovka notebook. We also propose a variant of Gowers’ trick for two
subsets, and give applications to products of large subsets in groups of Lie
type, improving some results of Larsen, Shalev and Tiep.

##  1 Introduction

Growth of subsets in finite simple groups has been an active area of research
in the recent years. In the most general setting the problem can be stated as
follows. Given two subsets $A,B$ of a finite group $G$, let

$AB=\\{ab\mid a\in A,\,b\in B\\}\text{ and }A^{-1}=\\{a^{-1}\mid a\in A\\}$

denote their product and inverse, respectively. We want to know how large
$|AB|$ is in comparison to $|A|$ and $|B|$, perhaps under some additional
conditions like $A=B$ and $A^{-1}=A$. Of course, for nonempty subsets $|AB|$
is always at least $\max\\{|A|,|B|\\}$ and at most $|A|\cdot|B|$, but in many
cases one can prove more.

One of such cases arises in the study of Babai’s conjecture on diameters of
Cayley graphs of finite simple groups. Recall that in [2] it is conjectured
that a connected Cayley graph of a nonabelian finite simple group $G$ has
polylogarithmic diameter. That is, there exist universal constants $C,c>0$
such that for any symmetric generating set $S\subseteq G$ we have $S^{n}=G$,
where $n\leq C\cdot(\log|G|)^{c}$ and $S^{n}=SS\cdots S$ denotes the $n$-fold
product set. In the breakthrough work of Helfgott [12] it was shown that for
$G=\mathrm{PSL}_{2}(p)$, where $p$ is a prime, generating sets exhibit three-
step growth, more precisely, we have $|S^{3}|\geq|S|^{1+\epsilon}$ or
$G=S^{3}$, where $\epsilon>0$ is a constant. It can be easily seen that this
property immediately implies the positive solution of Babai’s conjecture in
the case of $\mathrm{PSL}_{2}(p)$.

Similar growth theorems have been shown independently by Pyber, Szabó [23] and
Breuillard, Green, Tao [3] for groups of Lie type of bounded rank. It is known
that such growth results do not hold for groups of Lie type of unbounded rank
and for the alternating groups (see, for instance, [23, Example 77]), but if
one assumes the generating set $S$ to be _normal_ , i.e. $S$ is a union of
conjugacy classes of $G$, then the situation becomes much more benign.

Indeed, Babai’s conjecture in the case when the generating set is normal was
confirmed by Liebeck and Shalev in [18]. Two-step expansion of normal subsets
in finite simple groups was shown in [8] and improved in [20] for small
subsets. Results on expansion of two normal subsets include the fundamental
results of Shalev on products of conjugacy classes [24], including
applications to word maps and Thompson’s conjecture, see a recent survey [25]
for many generalizations and improvements that followed. Some expansion can be
obtained when $A$ is an arbitrary small subset of $G$ and $B$ is a nontrivial
normal subset, see [8, Proposition 5.2], [20, Theorem 1.3] and a recent result
by Dona, Maróti and Pyber [4].

In this work we will also focus on the expansion of normal subsets. A standard
strategy to study the product of normal subsets is to apply the Frobenius
formula, which expresses the product of two conjugacy classes in a finite
group in terms of irreducible ordinary characters, and then to utilize various
bounds on character values, such as the results on the Witten zeta function
(see, for example, [19]) and character ratio bounds (see [15] for a recent
strong bound). Here we choose a different strategy and instead appeal to some
classical results about expander graphs. Our main observation is that _the
character ratio_

$R(g)=\max_{\chi\neq 1_{G}}\left|\frac{\chi(g)}{\chi(1)}\right|,$

_where $g\in G$ and $\chi$ ranges over nontrivial irreducible characters of
$G$, controls the spectral expansion of the Cayley graph
$\operatorname{Cay}(G,g^{G})$ over $G$ with the connection set $g^{G}$._ We
formulate this observation more precisely in Lemma 1 of Section 2, but
intuitively it implies that the conjugacy class $g^{G}$ “expands” whenever
$R(g)$ is small enough. This often holds for conjugacy classes of finite
simple groups, which allows us to obtain growth results.

We note that this observation has been used before to show that normal Cayley
graphs over finite simple groups of Lie type are expanders, if the field of
definition is large enough [22, (3.10)]. Here we make one more step and show
that the expansion properties of Cayley graphs have some other group-
theoretical applications.

Our first result follows from the observation on character ratios together
with the well-known fact that spectral expansion implies vertex expansion (we
postpone the proofs of this and other results to Section 3).

######  Theorem 1.

Let $A,B$ be nonempty subsets of a finite group $G$ of order $n$, and assume
that $A$ is a normal subset. Define

$R=\min_{g\in A}R(g).$

Then

$|AB|\geq\frac{n}{1+R^{2}(n/|B|-1)}\geq\min\left\\{\frac{n}{2},\,\frac{|B|}{2R^{2}}\right\\}.$

Our main application throughout the text will be to the case when $G$ is a
finite simple group of Lie type over the field of order $q$, and $A$ is a
nontrivial normal subset. By Gluck’s bound [10], for $g\neq 1$ we have
$R(g)\leq C/\sqrt{q}$ for some universal constant $C$, so Theorem 1 implies
that either $|AB|\geq n/2$ or $|AB|\geq c\cdot q|B|$ for some constant $c>0$.
In fact, it follows from Gluck’s work that $R(g)\leq 19/20<1$ for any finite
simple group of Lie type, so we always get some mild expansion.

In [1, Corollary 2.5] Babai, Nikolov and Pyber proved that for arbitrary
nonempty subsets $A$ and $B$ we have either $|AB|\geq n/2$ or $|AB|\geq
m|A|/(2n)\cdot|B|$, where $m$ is the minimum degree of a nontrivial ordinary
character of $G$. In Section 4 we show that this result and some other results
from [1] are also a consequence of a (different) bound on spectral expansion
of Cayley graphs. In particular, Theorem 1 is an improvement to [1, Corollary
2.5] when $A$ is a single conjugacy class of $G$, since in that case $R$ is
precisely the spectral expansion of $\operatorname{Cay}(G,A)$.

In [7, Theorem 3.3] Gowers showed that if $A,B,C\subseteq G$ are such that
$|A|\cdot|B|\cdot|C|\geq|G|^{3}/m$, where $m$ is the minimum degree of a
nontrivial ordinary character of $G$, then $G=ABC$. This useful fact was later
called the “Gowers’ trick”, and was subsequently generalized to an arbitrary
number of sets in [1, Corollary 2.6] and recently improved for normal subsets
of groups of Lie type in [5]. Unfortunately these results do not yield a
nontrivial bound for two subsets. Our next result shows that Theorem 1 implies
such a bound.

######  Corollary 1 (“Gowers’ trick for two subsets”).

Let $A,B$ be subsets of a finite group $G$. If $|A|\cdot|B|\geq
R(g)^{2}\cdot|G|^{2}$ for $g\in G$, $g\neq 1$, then $AB\cap
g^{G}\neq\varnothing$. In particular, if $A$ and $B$ are normal subsets, then
$g^{G}\subseteq AB$.

In [16, Theorem A (iv)] Larsen, Shalev and Tiep prove that given a finite
simple group of Lie type $G$ of bounded rank, and two normal subsets
$A,B\subseteq G$ with $|A|,|B|\geq\epsilon|G|$, $\epsilon>0$, we have
$G\setminus\\{1\\}\subseteq AB$ if $|G|$ is sufficiently large (in terms of
$\epsilon$). Corollary 1 and Gluck’s bound imply that one does not have to
assume that $G$ has bounded rank, it is enough for the field of definition to
be sufficiently large.

######  Corollary 2.

There exists a universal constant $C>0$ such that the following is true. Let
$G$ be a finite simple group of Lie type over a field of order $q$, and let
$A$ and $B$ be normal subsets of $G$. If $|A|\cdot|B|\geq C/q\cdot|G|^{2}$,
then $G\setminus\\{1\\}\subseteq AB$.

Note that the proof of [16, Theorem A (iv)] is rather involved and requires
algebraic geometry, while our argument is elementary, aside from using Gluck’s
bound.

As an application of Corollary 2, we partially resolve a question of Pyber
from the Kourovka notebook. In [13, 20.74 (a)] he asks if given a nonabelian
finite simple group $G$ and a normal subset $A$ with $|A|>|G|/\log_{2}|G|$ and
$A^{-1}=A$ we have $A^{2}=G$ for $|G|$ large enough. In [13, 20.74 (b)] the
same question is posed in the particular case of $G=\mathrm{PSL}_{2}(p)$.
Pyber’s question was recently solved in the positive for alternating groups,
see [17, Theorem 4], and we settle it for finite simple groups of Lie type of
bounded rank.

######  Corollary 3.

Let $G$ be a finite simple group of Lie type of bounded rank, and let
$A\subseteq G$ be a normal subset satisfying $|A|>|G|/\log_{2}|G|$ and
$A^{-1}=A$. If $|G|$ is large enough in terms of the rank, then $A^{2}=G$.

Note that part (b) of Pyber’s question about $G=\mathrm{PSL}_{2}(p)$ follows
from the above corollary.

It is often important not only to decide when a given $g\in G$ lies in the
product of subsets $A,B\subseteq G$, but also to obtain an asymptotic formula
for the number of decompositions of the form $g=ab$, where $a\in A$ and $b\in
B$, see, for instance, [25, Question 1.4]. Let $P_{A,B}(g)$ denote the
probability that $g=ab$ where $a\in A$, $b\in B$ are chosen uniformly
independently at random, in other words

$P_{A,B}(g)=\frac{1}{|A|\cdot|B|}|\\{(a,b)\in A\times B\mid g=ab\\}|.$

Using the expander mixing lemma (Proposition 2) we can improve Corollary 1 and
show that $P_{A,B}(g)$ for nontrivial $g$ behaves close to a uniform
distribution.

######  Theorem 2.

Let $A,B$ be nonempty normal subsets of a finite group $G$. For $g\in G$ we
have

$\left|P_{A,B}(g)-\frac{1}{|G|}\right|<\frac{R(g)}{\sqrt{|A|\cdot|B|}}.$

Similarly to [16, Theorem A (iv)], in the case of groups of Lie type we can
also obtain an asymptotic formula for $P_{A,B}(g)$, when $|A|\geq\epsilon|G|$,
$|B|\geq\epsilon|G|$ for some $\epsilon>0$.

######  Corollary 4.

Let $G$ be a finite simple group of Lie type over the field of order $q$.
Assume that $A,B\subseteq G$ are normal subsets with $|A|\geq\epsilon|G|$,
$|B|\geq\epsilon|G|$ for some $\epsilon>0$. Then for any $g\in G$, $g\neq 1$
we have

$P_{A,B}(g)=\frac{1}{|G|}\left(1+O\left(\frac{1}{\epsilon\sqrt{q}}\right)\right).$

In the case when the normal subsets are images of group words and the group
has bounded rank, we obtain a version of [16, Theorem 8.1 (i)] with a more
explicit bound.

######  Corollary 5.

Let $G$ be a finite simple group of Lie type of bounded rank over the field of
order $q$. Let $w_{1},w_{2}$ be nonidentity group words. Then for any $g\in
G$, $g\neq 1$ we have

$P_{w_{1}(G),w_{2}(G)}(g)=\frac{1}{|G|}\left(1+O\left(\frac{1}{\sqrt{q}}\right)\right),$

where the implied constant in big-O depends on the rank of $G$ only.

Our final application of results on expanders concerns the growth of squares
of normal subsets in groups of Lie type. Gill, Pyber, Short and Szabó [8]
proved that given a nontrivial normal subset $A$ of a finite simple group $G$,
we either have $|A^{2}|\geq|A|^{1+\epsilon}$ or $A^{b}=G$ for some universal
constants $\epsilon>0$, $b\geq 1$. In the case of finite simple groups of Lie
type of bounded rank we can take $b=2$.

######  Theorem 3.

Let $G$ be a finite simple group of Lie type, and let $A$ be a nontrivial
normal subset of $G$. Then either $|A^{2}|\geq|A|^{1+\epsilon}$ or
$G\setminus\\{1\\}\subseteq A^{2}$, for some constant $\epsilon>0$ depending
on the rank of $G$ only.

Note that it is not always possible to replace the conclusion
$G\setminus\\{1\\}\subseteq A^{2}$ with $G=A^{2}$. Recall that an element of a
group is called _real_ if it is conjugate to its inverse. In the case of
$G=\mathrm{PSL}_{2}(q)$ one can easily show that every semisimple element is
real, so there exists a universal constant $C>0$ such that any normal subset
$A\subseteq G$ with $|A|\geq C\cdot q^{2}$ must contain a real element, and
hence $G=A^{2}$ follows. For $G=\mathrm{PSL}_{3}(q)$ the situation is
different. One can show that $G$ contains at most $O(q)$ conjugacy classes of
real elements (see [26, Table 2] or [9, Section 13]) and since every conjugacy
class in $G$ has size at most $O(q^{6})$, the group has at most $O(q^{7})$
real elements. On the other hand, $|G|=\frac{1}{\gcd(3,q-1)}q^{8}-O(q^{6})$,
in particular, $G$ contains at least $|G|(1-C/q)$ elements which are not real
for some universal constant $C>0$. Hence there exists a normal subset
$A\subseteq G$ of size at least $\frac{|G|}{2}(1-C/q)$ such that $1\not\in
A^{2}$; note that for any $\nu<1/2$ we can choose $A$ to be bigger than
$\nu|G|$ for $q$ large enough. The same example also shows why it is important
to exclude the identity element in Corollaries 2 and 4.

In general, the abundance of real elements in a group of Lie type depends on
the type of the group. In [6, Lemma 10] it is shown that in finite simple
groups of Lie type besides the groups of types $A_{n}$, ${}^{2}A_{n}$,
$D_{2n+1}$, ${}^{2}D_{2n+1}$, $E_{6}$ and ${}^{2}E_{6}$ every semisimple
element is real. By [11], a group of Lie type $G$ defined over a field of
order $q$ has at least $|G|(1-C/q)$ semisimple elements for some universal
constant $C>0$. It follows that in groups not of the six types above almost
any element is real, and hence we can strengthen Theorem 3 a bit in this case,
by showing that for $A\subseteq G$ we either have
$|A^{2}|\geq|A|^{1+\epsilon}$ or $G=A^{2}$.

The structure of the paper is as follows. In Section 2 we remind some basic
properties of expanders and prove the relation between character ratios and
spectral expansion of normal Cayley graphs (Lemma 1). In Section 3 we give the
proofs of our main results and in Section 4 we note how some other results on
expansion in finite simple groups fit into the framework of expander graphs.

##  2 Preliminaries

We follow [27, Chapter 4] for our exposition of expanders. Let $\Gamma=(V,E)$
be a directed graph (shortly, a _digraph_) with the vertex set $V$ and the arc
set $E\subseteq V\times V$. Suppose that $\Gamma$ is regular of valency $d$,
i.e. for every vertex $x\in V$ there are exactly $d$ arcs with the beginning
at $x$. Set $n=|V|$. Let $M$ denote the _random-walk matrix_ of the digraph,
that is, $M$ is a $n\times n$ matrix where $M_{x,y}$, $x,y\in V$, is equal to
$1/d$ if $(x,y)\in E$ and $0$ otherwise. Define

$\lambda(\Gamma)=\max_{x\perp u}\frac{\|xM\|}{\|x\|},$

where $u=(1/n,\dots,1/n)\in\mathbb{R}^{n}$ and $x\in\mathbb{R}^{n}$ ranges
over all vectors orthogonal to $u$. The norm $\|\cdot\|$ denotes the Euclidean
norm.

It can be easily seen that $1$ is always an eigenvalue of $M$ with an
eigenvector $u$. When $\Gamma$ is an undirected graph, $\lambda(\Gamma)$ is
the second largest eigenvalue of $M$ by absolute value, see [27, Lemma 2.55].
By [27, Problem 2.10 (2)], for an arbitrary digraph,
$\lambda(G)=\sqrt{\lambda_{2}}$, where $\lambda_{2}$ is the second largest
eigenvalue of $MM^{t}$; recall that this matrix is positive semidefinite. We
always have $\lambda(\Gamma)\leq 1$, and we call $\lambda(\Gamma)$ the
_spectral expansion_ of $\Gamma$. Note that in [27, Definition 4.5] spectral
expansion is defined in terms of the spectral gap $1-\lambda(\Gamma)$, but our
choice of terminology should not lead to any confusion (see the footnote to
[27, Definition 4.5]).

We say that a digraph $\Gamma$ is a $(N,a)$ _vertex expander_ , if for all
$A\subseteq V$ with $|A|\leq N$, the neighborhood $N(A)=\\{y\in V\mid(x,y)\in
E,\,\text{ for some }x\in A\\}$ satisfies $|N(A)|\geq a|A|$.

######  Proposition 1 (Spectral expansion implies vertex expansion [27,
Theorem 4.6]).

If $\Gamma$ is a regular digraph on $n$ vertices, then for all
$0\leq\alpha\leq 1$ the digraph $\Gamma$ is a $(\alpha
n,1/((1-\alpha)\lambda(\Gamma)^{2}+\alpha))$ vertex expander.

Given $A,B\subseteq V$, let $e(A,B)=|E\cap(A\times B)|$ denote the number of
arcs from $A$ to $B$.

######  Proposition 2 (Expander mixing lemma [27, Lemma 4.15]).

If $\Gamma=(V,E)$ is a regular digraph of valency $d$ on $n$ vertices, then
for all $A,B\subseteq V$ with $\alpha=|A|/n$, $\beta=|B|/n$ we have

$\left|\frac{e(A,B)}{dn}-\alpha\beta\right|\leq\lambda(\Gamma)\sqrt{\alpha(1-\alpha)\beta(1-\beta)}.$

For a finite group $G$ and a subset $S\subseteq G$, let
$\operatorname{Cay}(G,S)$ denote the _directed Cayley graph_ on $G$ with the
connection set $S$. The set of vertices is $G$, and we draw an arc from $g\in
G$ to $h\in G$ if and only if $g^{-1}h\in S$.

Let $\operatorname{Irr}(G)$ denote the set of irreducible ordinary characters
of a group $G$, and let $1_{G}$ denote the trivial character. Next result
shows that spectral expansion of normal Cayley graphs is controlled by
character ratios, cf. [22, (4.6)].

######  Lemma 1.

Let $G$ be a finite group with a normal subset $S\subseteq G$. Then

$\lambda(\operatorname{Cay}(G,S))\leq\max_{g\in S}R(g).$

Moreover, if $S$ is a conjugacy class, then we have an equality in the formula
above.

###### Proof.

Let $M$ denote the random-walk matrix of $\operatorname{Cay}(G,S)$. Define two
linear operators $\mathcal{M},\mathcal{M}^{*}:\mathbb{C}[G]\to\mathbb{C}[G]$
on the group algebra $\mathbb{C}[G]$ as follows:

$\mathcal{M}(h)=\frac{1}{|S|}\sum_{s\in S}s^{-1}h\,\text{ and
}\mathcal{M}^{*}(h)=\frac{1}{|S|}\sum_{s\in S}sh,$

for $h\in\mathbb{C}[G]$. It can be easily seen that the matrices of
$\mathcal{M}$ and $\mathcal{M}^{*}$ in the standard basis are $M$ and $M^{t}$
respectively. Since $S$ is a normal subset of $G$, the sums $\sum_{s\in
S}s^{-1}$ and $\sum_{s\in S}s$ lie in the center of the group algebra and
hence $\mathcal{M}(\mathcal{M}^{*}(h))=\mathcal{M}^{*}(\mathcal{M}(h))$ for
all $h\in\mathbb{C}[G]$. Therefore $MM^{t}=M^{t}M$, and hence the eigenvalues
of $MM^{t}$ are $|\lambda|^{2}$ where $\lambda$ runs through the eigenvalues
of $M$.

Set $R=\max_{g\in S}R(g)$. It is well known (see, for instance, [28, Theorem
1]) that eigenvalues of $M$ have the form

$\lambda_{\chi}=\frac{1}{\chi(1)|S|}\sum_{g\in S}\chi(g)$

where $\chi$ ranges over $\operatorname{Irr}(G)$. The largest eigenvalue is
$\lambda_{1_{G}}=1$, so for $\chi\neq 1_{G}$ we have

$|\lambda_{\chi}|\leq\frac{1}{\chi(1)|S|}\sum_{g\in S}|\chi(g)|\leq\max_{g\in
S}\left|\frac{\chi(g)}{\chi(1)}\right|\sum_{g\in S}\frac{1}{|S|}\leq\max_{g\in
S}R(g)=R.$

Now, it follows that the second largest eigenvalue $\lambda_{2}$ of $MM^{t}$
is at most $R^{2}$. Hence
$\lambda(\operatorname{Cay}(G,S))=\sqrt{\lambda_{2}}\leq R$ as claimed.

If $S$ is a conjugacy class $g^{G}$, then $\lambda_{\chi}=\chi(g)/\chi(1)$ and
hence $\lambda(\operatorname{Cay}(G,S))=R$ as wanted. ∎

The following is the main character ratio bound that we will be using.

######  Proposition 3 (Gluck’s bound [10]).

Let $G$ be a finite simple group of Lie type over the field of order $q$. For
any $\chi\in\operatorname{Irr}(G)$, $\chi\neq 1_{G}$, and any $x\in G$, $x\neq
1$ we have $|\chi(x)/\chi(1)|\leq C/\sqrt{q}$ for a universal constant $C>0$.

##  3 Proofs of main results

_Proof of Theorem 1._ Let $g\in A$ be such that $R=R(g)$. Set
$\Gamma=\operatorname{Cay}(G,g^{G})$. We have $|AB|\geq|Bg^{G}|\geq|N(B)|$,
where $N(B)$ is the neighborhood of $B$ in $\Gamma$. By Proposition 1, the
digraph $\Gamma$ is $(\alpha
n,1/((1-\alpha)\lambda(\Gamma)^{2}+\alpha))$-expander for any $0\leq\alpha\leq
1$. Take $\alpha=|B|/n$ and recall that $\lambda(\Gamma)\leq R$ by Lemma 1.
Now we have

$|AB|\geq|N(B)|\geq\frac{|B|}{(1-|B|/n)R^{2}+|B|/n}=\frac{n}{1+R^{2}(n/|B|-1)}.$

It is easy to see that the right hand side is at least
$\min\\{n/2,\,|B|/(2R^{2})\\}$, so the claim follows. ∎

_Proof of Corollary 1._ We apply Theorem 1 to sets $g^{G}$ and $B^{-1}$. Let
$n=|G|$. Recalling that $|B|=|B^{-1}|$ we obtain

$|g^{G}\cdot B^{-1}|\geq\frac{n}{1+R(g)^{2}(n/|B|-1)}>n(1-R(g)^{2}(n/|B|-1)).$

By assumption, $|A|\cdot|B|\geq n^{2}R(g)^{2}\geq nR(g)^{2}(n-|B|)$, hence

$|g^{G}\cdot B^{-1}|>n(1-R(g)^{2}(n/|B|-1))\geq n-|A|.$

Therefore $|g^{G}\cdot B^{-1}|+|A|>n$, and so $g^{G}\cdot B^{-1}\cap
A\neq\varnothing$. It follows that $AB\cap g^{G}\neq\varnothing$. If $A$ and
$B$ are normal, then so is $AB$ and the final claim follows. ∎

_Proof of Corollary 2._ We apply Corollary 1 and note that for any $g\in
G\setminus\\{1\\}$ we have $R(g)\leq C/\sqrt{q}$ for some universal constant
$C$ by Proposition 3. ∎

_Proof of Corollary 3._ Recall that the group $G$ of Lie type of bounded rank
defined over the field of order $q$ has order at most $q^{c}$ for some
constant $c$ depending on the rank only. Corollary 2 implies that if
$A\subseteq G$ with $A^{-1}=A$ satisfies $|A|\geq\sqrt{C/q}\cdot|G|$, then
$A^{2}=G$. Since $\log_{2}|G|\leq c\log_{2}q\leq\sqrt{q/C}$ for $q$ large
enough, we have $|A|>|G|/\log_{2}|G|\geq\sqrt{C/q}\cdot|G|$, so the conditions
of Corollary 2 are satisfied and $A^{2}=G$ as claimed. ∎

_Proof of Theorem 2._ Consider $\Gamma=\operatorname{Cay}(G,g^{G})$ for $g\in
G$, and recall that $\lambda(\Gamma)\leq R(g)$ by Lemma 1. By Proposition 2
applied to $\Gamma$ and subsets $A^{-1}$ and $B$ we get

$\left|\frac{e(A^{-1},B)}{|g^{G}|\cdot|G|}-\frac{|A^{-1}|\cdot|B|}{|G|^{2}}\right|<\frac{R(g)}{|G|}\sqrt{|A^{-1}|\cdot|B|}.$

Notice that there is an arc between an element $a\in A^{-1}$ and $b\in B$ if
and only if $a^{-1}b\in g^{G}$. In particular, $e(A^{-1},B)=|\\{(a,b)\in
A\times B\mid ab\in g^{G}\\}|$. Since $A$ and $B$ are normal subsets, and
since all elements of $g^{G}$ are conjugate, the number of decompositions of
the form $x=ab$, $a\in A$, $b\in B$, does not depend on the choice of $x\in
g^{G}$. Therefore

$P_{A,B}(g)=\frac{e(A^{-1},B)}{|g^{G}|\cdot|A|\cdot|B|}.$

We derive

$\left|\frac{P_{A,B}(g)\cdot|A^{-1}|\cdot|B|}{|G|}-\frac{|A|\cdot|B|}{|G|^{2}}\right|<\frac{R(g)}{|G|}\sqrt{|A^{-1}|\cdot|B|},$

and the theorem follows after dividing the inequality by $|A|\cdot|B|/|G|$ and
replacing $|A^{-1}|$ by $|A|$. ∎

_Proof of Corollary 4._ Follows directly from Theorem 2 and Proposition 3. ∎

_Proof of Corollary 5._ By [14], there exists a constant $\epsilon>0$
depending on the rank of $G$ only, such that $|w_{i}(G)|\geq\epsilon|G|$,
$i=1,2$. Now the claim follows from Corollary 4. ∎

The proof of Theorem 3 will follow from a more precise result.

######  Lemma 2.

Let $G$ be a nontrivial finite group, and define

$R=\max_{g\in G\setminus\\{1\\}}R(g).$

Let $A$ be a nontrivial normal subset of $G$. If $|A|\geq R|G|$, then
$G\setminus\\{1\\}\subseteq A^{2}$. If $|A|<R|G|$, then $|A^{2}|\geq|A|/(2R)$.

###### Proof.

If $|A|\geq R|G|$, then the claim follows from Corollary 1. Assume that
$|A|<R|G|$. Define $R^{\prime}=\min_{g\in A}R(g)$, and observe that
$R^{\prime}\leq R$. By Theorem 1, we have

$|A^{2}|\geq\frac{|G|}{1+R^{\prime
2}(|G|/|A|-1)}\geq\frac{|A|}{|A|/|G|+R^{2}(1-|A|/|G|)}\geq\frac{|A|}{R+R^{2}}\geq\frac{|A|}{2R},$

and the claim follows. ∎

_Proof of Theorem 3._ Let $G$ be a finite simple group of Lie type of rank $r$
defined over the field of order $q$. We apply Lemma 2. If $|A|\geq R|G|$, then
$G\setminus\\{1\\}\subseteq A^{2}$, so we may assume that $|A|<R|G|$. Recall
that $R\leq C/\sqrt{q}$ for some universal constant $C$ by Proposition 3, so
$|A^{2}|\geq|A|/(2R)\geq|A|\sqrt{q}/(2C)$. Now, $|G|\leq q^{cr^{2}}$ for some
universal constant $c$, hence
$\sqrt{q}/(2C)\geq|G|^{\epsilon}\geq|A|^{\epsilon}$, where
$\epsilon=c^{\prime}/r^{2}$ for some universal constant $c^{\prime}>0$.
Therefore $|A^{2}|\geq|A|^{1+\epsilon}$ and the claim follows. ∎

##  4 Final remarks

We have often compared our results with those obtained by Babai, Nikolov and
Pyber in [1], and it turns out that the proof method employed there is also
related to expander graphs. To state the main result of [1], recall that given
a finite group $G$, a function $X:G\to\mathbb{R}_{\geq 0}$ is called a
_probability distribution_ on $G$, if $\sum_{g\in G}X(g)=1$. Given two
distributions $X,Y:G\to\mathbb{R}_{\geq 0}$, their _convolution_
$X*Y:G\to\mathbb{R}_{\geq 0}$ is also a distribution defined as
$(X*Y)(h)=\sum_{g\in G}X(g)Y(g^{-1}h)$. For a function $f:G\to\mathbb{R}$, let
$\|f\|=\sqrt{\sum_{g\in G}|f(g)|^{2}}$ denote the $\ell_{2}$-norm. Now, if
$U:G\to\mathbb{R}_{\geq 0}$ is the uniform distribution on $G$, then [1,
Theorem 2.1] gives the following inequality:

$\|X*Y-U\|\leq\sqrt{n/m}\cdot\|X-U\|\cdot\|Y-U\|,$

where $n$ is the order of $G$ and $m$ is the minimum degree of a nontrivial
ordinary representation of $G$. All the expansion results in [1] follow from
this convolution inequality, for instance, to obtain two-step expansion [1,
Corollary 2.4] (which we improve for normal subsets in Theorem 1), inequality
$(*)$ is applied to distributions concentrated on subsets $A,B\subseteq G$.

We claim that this inequality is equivalent to a spectral expansion bound for
a certain weighted Cayley digraph. Keeping the same notation for the group and
the distributions, let $\Gamma$ be a complete Cayley digraph on $G$, where an
arc between $x\in G$ and $y\in G$ has weight $Y(x^{-1}y)$. The random-walk
matrix $M$ of this digraph is a $n\times n$ matrix with entries
$M_{x,y}=Y(x^{-1}y)$; here we identify indices used with the group elements.
By [27, Section 4.1.2], the spectral expansion of this digraph is equal to

$\lambda(\Gamma)=\max_{\pi}\frac{\|\pi M-u\|}{\|\pi-u\|},$

where the vector $u=(1/n,\dots,1/n)\in\mathbb{R}^{n}$ corresponds to the
uniform distribution, and $\pi$ ranges over all vectors from $\mathbb{R}_{\geq
0}^{n}$ with $\sum_{g}\pi_{g}=1$, in other words, $\pi$ ranges over arbitrary
probability distributions on $G$. Now, if one takes $\pi_{g}=X(g)$, one can
easily check that $(\pi M)_{g}=(X*Y)(g)$. Therefore $\|\pi M-u\|=\|X*Y-U\|$
and $\|\pi-u\|=\|X-U\|$. Plugging this into the formula for $\lambda(\Gamma)$
above and recalling that $X$ in $(*)$ is an arbitrary distribution, we obtain
the following bound for the spectral expansion of $\Gamma$:

$\lambda(\Gamma)\leq\sqrt{n/m}\cdot\|Y-U\|.$

Since $X$ in $(*)$ was arbitrary, we can see that $(*)$ and $(*^{\prime})$
are, in fact, equivalent.

We note that the proof of [1, Theorem 2.1] makes use of a bound on a second
largest eigenvalue of a certain matrix, see [1, Lemma 5.7], so the relation
between $(*)$ and spectral expansion is rather natural, albeit slightly hidden
by the formulation of the main result of [1].

To illustrate our point even further, we provide an alternative proof of two-
step expansion obtained in [1, Corollary 2.5], using the spectral expansion.

######  Proposition 4 (Alternative proof of [1, Corollary 2.5]).

Let $A$ and $B$ be nonempty subsets of a finite group $G$ of order $n$, and
let $m$ be the minimum degree of a nontrivial ordinary representation of $G$.
Then

$|AB|>\frac{n}{1+\frac{n^{2}}{m|A||B|}}\geq\min\left\\{\frac{n}{2},\,\frac{m|A||B|}{2n}\right\\}.$

###### Proof.

We follow the proof of Theorem 1. Set $\Gamma=\operatorname{Cay}(G,B)$. The
digraph $\Gamma$ is a $(\alpha
n,(1-\alpha)\lambda(\Gamma)^{2}+\alpha)$-expander for $\alpha=|A|/n$. Hence

$|AB|\geq|N(A)|\geq\frac{|A|}{(1-|A|/n)\lambda(\Gamma)^{2}+|A|/n}=\frac{n}{(n/|A|-1)\lambda(\Gamma)^{2}+1}.$

Let $Y:G\to\mathbb{R}_{\geq 0}$ be a distribution on $G$ defined as
$Y(g)=1/|B|$ if $g\in B$ and $Y(g)=0$ otherwise. By $(*^{\prime})$, we have

$\lambda(\Gamma)\leq\sqrt{n/m}\cdot\|Y-U\|=\sqrt{n/m}\cdot\sqrt{1/|B|-1/n}\leq\sqrt{\frac{n}{m|B|}}.$

Hence

$|AB|\geq\frac{n}{(n/|A|-1)\lambda(\Gamma)^{2}+1}>\frac{n}{n/|A|\cdot
n/(m|B|)+1},$

as claimed. The inequality with the minimum follows trivially. ∎

Gowers’ trick for three subsets follows from this result, so it is also a
corollary of expansion properties of Cayley graphs.

Of course, we do not claim that the convolution inequality $(*)$ is rendered
unnecessary by the more popular machinery of spectral expansion of Cayley
graphs. For instance, the fact that $(*)$ works for arbitrary distributions
$X$ and $Y$ allows us to iterate the inequality and obtain bounds on
convolutions of many distributions, see [1, Corollary 2.2 and Corollary 2.3].
We also note that although Lemma 1 gives a precise formula for spectral
expansion with respect to a conjugacy class, for arbitrary normal subsets the
bound $(*^{\prime})$ can turn out to be better.

We finally mention that some recent strong results on expansion of normal
subsets are formulated in terms of convolutions, see [21, Theorem 1.4].

##  5 Acknowledgements

The author would like to thank prof. A.A. Gal’t, L. Pyber and A.V Vasil’ev for
fruitful discussions improving the paper considerably.

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_Saveliy V. Skresanov_

_Sobolev Institute of Mathematics, 4 Acad. Koptyug avenue, Novosibirsk,
Russia_

_Email address<EMAIL_ADDRESS>


# NEBULAR C IV $\lambda 1550$ IMAGING OF THE METAL-POOR STARBURST MRK 71:
DIRECT EVIDENCE OF CATASTROPHIC COOLING

M. S. Oey University of Michigan, Department of Astronomy, 1085 South
University Ave., Ann Arbor, MI 48109-1107, USA  Amit N. Sawant University of
Michigan, Department of Astronomy, 1085 South University Ave., Ann Arbor, MI
48109-1107, USA  Present address: Institute of Mathematics, EPFL SB MATH, MA
C2 647, Station 8, CH-1015 Lausanne, Switzerland  Ashkbiz Danehkar  Eureka
Scientific, 2452 Delmer Street, Suite 100, Oakland, CA 94602-3017, USA  Sergiy
Silich  Instituto Nacional de Astrofísica, Óptica y Electr’onica (INAOE), AP
51, 72000 Puebla, México  Linda J. Smith  Space Telescope Science Institute,
3700 San Martin Dr., Baltimore, MD 21218, USA  Jens Melinder  Stockholm
University, Department of Astronomy and Oskar Klein Centre, AlbaNova
University Centre, SE-10691 Stockholm, Sweden  Claus Leitherer  Space
Telescope Science Institute, 3700 San Martin Dr., Baltimore, MD 21218, USA
Matthew Hayes  Stockholm University, Department of Astronomy and Oskar Klein
Centre, AlbaNova University Centre, SE-10691 Stockholm, Sweden  Anne E. Jaskot
Williams College, Department of Astronomy, Williamstown, MA 01267, USA
Daniela Calzetti  Department of Astronomy, University of Massachusetts,
Amherst, MA 01003, USA  You-Hua Chu  Institute of Astronomy and Astrophysics,
Academia Sinica (ASIAA), No. 1, Sec. 4, Roosevelt Road, Taipei 10617, Taiwan
Bethan L. James  Space Telescope Science Institute, 3700 San Martin Dr.,
Baltimore, MD 21218, USA  Göran Östlin  Stockholm University, Department of
Astronomy and Oskar Klein Centre, AlbaNova University Centre, SE-10691
Stockholm, Sweden

(Received September 1, 2023; Revised October 17, 2023; Accepted October 18,
2023)

###### Abstract

We use the Hubble Space Telescope ACS camera to obtain the first spatially
resolved, nebular imaging in the light of C iv $\lambda\lambda 1548,1551$ by
using the F150LP and F165LP filters. These observations of the local starburst
Mrk 71 in NGC 2366 show emission apparently originating within the interior
cavity around the dominant super star cluster (SSC), Knot A. Together with
imaging in He ii $\lambda 4686$ and supporting STIS FUV spectroscopy, the
morphology and intensity of the C iv nebular surface brightness and the C
iv/He ii ratio map provide direct evidence that the mechanical feedback is
likely dominated by catastrophic radiative cooling, which strongly disrupts
adiabatic superbubble evolution. The implied extreme mass loading and low
kinetic efficiency of the cluster wind are reasonably consistent with the wind
energy budget, which is probably enhanced by radiation pressure. In contrast,
the Knot B SSC lies within a well-defined superbubble with associated soft
X-rays and He ii $\lambda 1640$ emission, which are signatures of adiabatic,
energy-driven feedback from a supernova-driven outflow. This system lacks
clear evidence of C iv from the limb-brightened shell, as expected for this
model, but the observations may not be deep enough to confirm its presence. We
also detect a small C iv-emitting object that is likely an embedded compact H
ii region. Its C iv emission may indicate the presence of very massive stars
($>100\ \rm\,M_{\odot}$) or strongly pressure-confined stellar feedback.

starburst galaxies — galaxy winds — galaxy evolution — emission-line galaxies
— stellar feedback — young massive clusters — superbubbles — H ii regions —
ultraviolet photometry — direct imaging

††journal: ApJ Letters††facilities: HST(ACS, STIS)††software:
DrizzlePac(Hoffmann et al., 2021)

##  1 Introduction

Massive star feedback encompasses energetic processes that heat gas to
temperatures above $10^{4}$ K. OB stars and their hot, blue descendants, as
well as high-mass X-ray binaries, photoionize gas into this regime, and shock-
heating by supernovae and stellar winds drive temperatures up to $10^{6}$ to
$10^{8}$ K. C iv $\lambda\lambda 1548,1551$ (hereafter “C iv $\lambda 1550$”)
is ubiquitous in the interstellar medium of star-forming galaxies (e.g.,
Savage, 1984; Savage et al., 2001; Wang & Yao, 2005), where it is believed to
originate in conductive interfaces between hot ($>10^{6}$) K gas and cooler
ISM phases (e.g., McCray & Snow, 1979); this species is also a prominent
P-Cygni emission line arising in hot star winds. C iv $\lambda 1550$ traces
ionization energies above 47.9 eV, and for recombination, above 64.5 eV.
Nebular C iv is therefore only rarely seen, and is associated more with
planetary nebulae rather than ordinary H ii regions (e.g., Aller et al., 1981;
Harrington et al., 1982).

However, C iv $\lambda 1550$ emission does appear in a number of extreme
starbursts, both locally (Mingozzi et al., 2022; Berg et al., 2019; Senchyna
et al., 2019) and at high redshift, where it can be prominent (Stark et al.,
2015; Amorín et al., 2017; Senchyna et al., 2022). In these objects, it is
generally thought to be nebular, although its origin is not well understood.
It could be a signature of photoionization by unusually hot stars like Wolf-
Rayet (WR) stars or rapidly rotating stars, or by high-mass X-ray binaries
(HMXBs); or it could be due to mechanical feedback, whether from direct
collisional ionization and conductive interfaces to adiabatic heating zones
(e.g., Chu et al., 1994), or from radiative, catastrophic cooling flows (Gray
et al., 2019) that disrupt adiabatic conditions. The origin of C iv emission
is of particular cosmological interest (Senchyna et al., 2022) when linked to
low metallicites. This nebular line implies higher ionization parameters than
is normally seen in H ii regions, and different mechanisms have been proposed
to explain its presence in intense metal-poor starbursts. Under these
conditions stars are more compact, with faster stellar rotation, both of which
increase the effective temperatures. Low metallicity is also linked to
stronger interaction in close binaries, which promotes the formation of WR
stars and fast rotators by binary mass transfer, as well as the creation of
HMXBs.

Nebular C iv imaging of resolved, local objects would therefore provide an
important and revealing diagnostic for these different scenarios. But since
these $\lambda\lambda 1548,1551$ resonance lines are in the far ultraviolet,
it has generally been inaccessible for such targets. However, the Solar Blind
Channel (SBC) of the Advanced Camera for Surveys (ACS) aboard the Hubble Space
Telescope (HST) offers a long-pass filter set, F150LP and F165LP, that is
capable of imaging in the light of C iv $\lambda 1550$. The net transmission
in these filters is shown by Hayes et al. (2016), who used a similar filter
pair to successfully carry out imaging of the starburst galaxy SDSS
J115630.63+500822.1 in O vi $\lambda\lambda 1032,1038$. In this Letter, we
report the first spatially resolved imaging of C iv nebular emission, which
was carried out with the ACS/SBC.

Figure 1:  Top: Three-color HST/WFC3 image of Mrk 71 in F373N ([O ii]
$\lambda$3727), F502N ([O iii] $\lambda$5007), and F469N (He ii $\lambda$4686)
corresponding to red, green, and blue, respectively. These archive data were
obtained by James et al. (2016) (GO-13041; colors not to scale). Bottom: Our
new, total combined F150LP image, in units of $10^{-16}\ \rm erg\ s^{-1}\
cm^{-2}$, not corrected for reddening. Most of the diffuse emission seen in
this image is due to scattered starlight. Knots A and B are separated by
5.0$\arcsec$ (83 pc).

Our target is the local starburst complex Mrk 71 in the nearby Magellanic
irregular galaxy NGC 2366. This system is of intense interest since it is a
remarkable analog of extreme Green Pea galaxies (Micheva et al., 2017), which
are the only known class of local Lyman-continuum emitting galaxies (e.g.,
Izotov et al., 2018; Flury et al., 2022). At a distance of only 3.4 Mpc
(Tolstoy et al., 1995), Mrk 71 is close enough to resolve individual stars. It
is also a metal-poor system, having $12+\log(\rm O/H)=7.89$ (Izotov et al.,
1997; Chen et al., 2023), or about $0.16\rm\,Z_{\odot}$. The nature of
feedback changes dramatically at this low metallicity. In addition to hotter
stellar photoionizing sources described above, mechanical feedback is much
weaker (Jecmen & Oey, 2023) since at low metallicity, supernovae occur mostly
at the lower-mass range of core collapse progenitors (Patton & Sukhbold, 2020;
O’Connor & Ott, 2011; Heger et al., 2003) and stellar winds are dramatically
weaker (e.g., Vink, 2022; Ramachandran et al., 2019; Björklund et al., 2023).
Mrk 71 provides an outstanding template for metal-poor feedback processes
because it hosts both a young super star cluster (SSC) driving an extreme
ionization parameter (Knot A), and a second, more evolved, SSC (Knot B) that
has generated a mature superbubble system (Figure 1). Our C iv imaging yields
critical, diagnostic insight on both of these subsystems.

##  2 C iv imaging observations

We obtained Cycle 28 ACS/SBC imaging observations of Mrk 71 (GO-16261; PI Oey)
in F150LP and F165LP during 2020 Oct 28 – Nov 01, using the LODARK aperture.
The target was observed in F150LP for 4 $\times 1480$ s plus 4 $\times 1497$
s, yielding a total of 11,908 s. In F165LP, the total exposure was $8\times
1480$ s plus $8\times 1497$ s plus 3054 s, or 26,870 s total. We combined the
frames in each of the two bands using the STScI DrizzlePac software. The world
coordinate systems of the images were first aligned to an accuracy of 0.15 –
0.20 pixels, as limited by the non-gaussian point-spread function (PSF; see
Avila & Chiaberge, 2016). We then drizzled the 8 F150LP frames together,
obtaining a pixel scale of $0\arcsec.025$ pixel-1. The 17 F165LP images were
drizzled as a separate set, using the combined, drizzled F150LP image as the
reference. PSF matching was carried out using the Photutils package,
convolving both drizzled images with a combined PSF. The final combined F150LP
image is shown in the bottom panel of Figure 1. Extensive, diffuse emission is
seen throughout the region; however, most of this extended emission
corresponds to scattered starlight as noted by Drissen et al. (2000), and we
show below that it is removed by continuum subtraction.

Figure 2:  STIS FUV long-slit spectrum across Knots A and B. The nebular
emission features of C iv $\lambda 1550$, He ii $\lambda 1640$ and Oiii]
$\lambda\lambda 1661,1666$ are marked.

Figure 3: The top and bottom panels show shallow and deep logarithmic
contrasts, respectively, of the continuum-subtracted C iv $\lambda\lambda
1548,1551$ image. WR stars, LBV-V1, and an IR-bright “hot spot” (Drissen et
al., 1997, 2000) are identified. The circle around Knot A indicates the
$0.8\arcsec$-radius aperture used to measure the C iv nebular flux.

Our HST program also obtained a deep STIS long-slit spectrum across Knots A
and B, using the G140L grating which includes the C iv spectral region (Figure
2). These data are reduced and calibrated in our study of the stellar
population in Knot A (Smith et al., 2023). We use this spectrum to determine
the scale factor for the F165LP image to match the depth in F150LP at C iv,
thereby flux-calibrating the continuum-subtracted C iv image. This is carried
out by matching the observed diffuse, nebular C iv flux in two regions with
$0.\arcsec 5$ length along the $0.\arcsec 2$-wide slit on both sides adjacent
to Knot A. The resulting scale factor is consistent with the results obtained
using the method developed by Sawant et al. (2021) based on maximizing the
modal bin fraction. Shallow and deep versions of the final continuum-
subtracted image are shown in Figure 3. We note that the He ii $\lambda 1640$
emission line falls within the lower throughput regimes of both the F150LP and
F165LP filters; the transmission curves and F165LP scale factor imply that the
level of remaining He ii $\lambda 1640$ emission in the continuum-subtracted
image is on the order of 2% of the signal. Below, we specifically compare to
archive He ii $\lambda 4686$ emission.

Many stars are seen in the deep panel of Figure 3. Since, as described above,
the continuum subtraction is applied to the C iv emission and is independent
of the stellar data, these stars are mostly strong FUV emitters, and are not
necessarily C iv emitters. This is discussed further in Section 4. The
detection of stars is also enhanced by our deep continuum imaging, which
generates r.m.s. noise in the background of only $\sim 0.1$%. Diffuse emission
is also apparently seen around both Knots A and B in Figure 3. However,
inspection of the STIS spectrum in Figure 2 shows that only the diffuse
emission around Knot A is real. We believe that the spurious emission around
Knot B is caused by the SBC PSF. While its FWHM is $0.02\arcsec$ and its 50%
encircled energy radius is 0.075$\arcsec$ (Avila & Chiaberge, 2016), it has
broad, faint wings that appear to be imperfectly continuum subtracted and thus
may cause residual emission around the brightest stars. The effect extends out
to about $\sim 0\arcsec.25$, as can be seen around, e.g., LBV-V1 (Figure 3).
Knot B is 4.6$\times$ brighter than Knot A within an aperture of 0.35$\arcsec$
radius, and it also has many more UV-bright stars in a more spatially extended
configuration, whereas Knot A does not have many such stars in its immediate
vicinity.

##  3 Knot A: Catastrophic Cooling

The Knot A SSC strongly dominates the luminosity (e.g., González-Delgado et
al., 1994; Drissen et al., 2000) and Green Pea-like ionization parameter
($\log U=-2$; James et al., 2016) of the Mrk 71 complex. Its estimated mass is
$\sim 1.4\times 10^{5}\ \rm\,M_{\odot}$ based on the H$\alpha$ luminosity
(Micheva et al., 2017) and its age is $1\pm 1$ Myr based on stellar population
synthesis (Smith et al., 2023). The SSC is still substantially enshrouded, but
stellar features implying the presence of VMS stars are detected (Smith et
al., 2023).

Nebular C iv $\lambda 1550$ emission can originate from systems with weak
mechanical feedback via strong radiative cooling and/or high-energy
photoionization of dense gas retained near the parent SSC. Or, C iv can be
emitted from a conductive interface between a cool, dense shell and interior
hot gas generated by strong, energy-driven mechanical feedback. Thus, these
two scenarios produce contrasting morphologies in nebular C iv: interior
emission for weak mechanical feedback versus shell emission for strong
mechanical feedback (e.g., Danehkar et al., 2022, 2021; Gray et al., 2019), as
demonstrated further below.

Figure 4:  Outcomes for the modeled parameter space following Danehkar et al.
(2021, see text), where AB = adiabatic bubble, AP = pressure-confined
adiabatic bubble, AW = adiabatic wind only, NW = no wind, CB = partially
cooling bubble

, CC = full catastrophic cooling, CP = cooling pressure-confined, MC =
momentum conserving only. The code fails for the NW and MC cases since these
do not generate energy-driven models. The color scale shows the temperature of
the hot bubble region (Figures 5 and 6) relative to the value expected for
simple adiabatic expansion. See Danehkar et al. (2021) for a full explanation
and discussion of these categories.

Figure 3 shows diffuse C iv emission within a $sim15$-pc radius of the Knot A
SSC, assuming a distance of 3.4 Mpc (Tolstoy et al., 1995). This region is
coincident with the boundary of the dense gas to the west and south (Figure 1)
that has been identified as a cavity or shell created by mechanical feedback
from the SSC (Komarova et al., 2021; Oey et al., 2017). Observations of the
nebular and molecular gas kinematics for this region show that the shell
expansion velocity is only $\sim 5-10\ {\rm\,km\,s^{-1}}$ (Komarova et al.,
2021; Micheva et al., 2019; Oey et al., 2017). This localized offset in the
systemic velocity is a separate component from the faint, broad emission-line
wings that are a much more spatially extended feature discussed in detail by
Komarova et al. (2021). For the observed parameters associated with Knot A,
this local shell expansion velocity is consistent with momentum-conserving,
non-adiabatic expansion and thus this system has been suggested (Komarova et
al., 2021; Oey et al., 2017) to be an example of metal-poor feedback where
superwinds are suppressed (e.g., Jecmen & Oey, 2023).

Weak, dense superwinds can experience strong, radiative cooling that quenches
the energy-driven, adiabatic outflow (e.g., Silich et al., 2004; Krumholz &
Matzner, 2009; Lochhaas et al., 2021), and furthermore, weak winds may be
suppressed within the cluster itself, such that the individual stellar wind
bubbles fail to merge into a coherent outflow (Silich & Tenorio-Tagle, 2018;
Yadav et al., 2017). In the last case, an expanding cavity is formed by
photoionization and radiation pressure from the SSC. We refer to both of these
scenarios that disrupt adiabatic evolution as “catastrophic” cooling.
Following Danehkar et al. (2021), we adopt a criterion that the driving
superwind has dropped to a temperature $<75\%$ of the adiabatic value. Another
possibility is that energy-driven feedback may exist but may be pressure-
confined (Oey & Garcia‐Segura, 2004; Silich et al., 2007). Thus, we note that
while the shell velocity and parameter space for this object are more
suggestive of momentum-conserving evolution (e.g., Komarova et al., 2021), the
kinematics alone are insufficient to distinguish between an energy-driven,
adiabatic and non-adiabatic, catastrophic cooling regime. This underscores the
importance of ions like C iv as a diagnostic of the interior and shell
temperature structure.

Figure 5:  Models of volume emissivity ($\rm erg\ s^{-1}\ cm^{-3}$) for the
shown emission lines as a function of radius, for a conventional adiabatic
model (AB) generated with $V=1000\ {\rm\,km\,s^{-1}}$,
$\log\dot{M}=-3~{}/\rm\rm\,M_{\odot}\ yr^{-1}$, and $n=1000\ \rm cm^{-3}$ from
the grid in Figure 4. We also show the radial profiles for the temperature and
density of this model in the left panels. The dotted lines correspond to the
analytic relations $n\sim r^{-2}$ and $T\sim r^{-4/3}$ for an adiabatic,
freely expanding wind. The bottom emissivity panels show calculations for
combined collisional and photoionization (CPI) that are predicted for this
model. As a comparison, the top panels show models for pure photoionization
(PI) of the same density distribution, which would be largely isothermal on
the order of $10^{4}$ K (Danehkar et al., 2021).  Figure 6:  The same as
Figure 5, but for a catastrophic cooling model (CB) with $V=500\
{\rm\,km\,s^{-1}}$, $\log\dot{M}=-2~{}/\rm\rm\,M_{\odot}\ yr^{-1}$, and
$n=500\ \rm cm^{-3}$. The hot bubble interior is now much cooler than for the
AB model in Figure 5 and occupies a much smaller fractional volume interior to
the shell.

Following Danehkar et al. (2021), we calculate a grid of models for energy-
driven feedback using the Maihem non-equilibrium ionization code (Gray et al.,
2019), covering a range of parameters similar to those inferred for the Knot A
SSC. We assume a cluster radius of 1 pc (Micheva et al., 2017), mass $1\times
10^{5}\ \rm\,M_{\odot}$ with a Salpeter IMF having stellar mass range
$0.5-150\ \rm\,M_{\odot}$, age 1 Myr, and metallicity $0.1\ \rm\,Z_{\odot}$.
The effective wind velocity is modeled in the range $V=250-2000\
{\rm\,km\,s^{-1}}$, its effective mass-loss rate is in the range
$\log\dot{M}=-2$ to $-4~{}/~{}\rm\rm\,M_{\odot}\ yr^{-1}$, and the ambient
density is $n=100-1000\ \rm cm^{-3}$. The modeled ranges for $V$ and $\dot{M}$
are based on those expected for the combined stellar winds from the SSC and
extend to values that generate suppressed superwind conditions similar to
those in (Danehkar et al., 2021). Figure 4 shows the distribution of feedback
outcomes, ranging from fully adiabatic bubbles to fully momentum-conserving
outflows.

In Figures 5 and 6, we show radial emissivities calculated following Danehkar
et al. (2022) using Cloudy (Ferland et al., 2017) for two of the grid models
that are broadly similar to the observed shell size, and which offer
contrasting interpretations of adiabatic and catastrophic cooling,
respectively. The model generated with $V=1000\ {\rm\,km\,s^{-1}}$,
$\log\dot{M}=-3/\rm\rm\,M_{\odot}\ yr^{-1}$, and $n=1000\ \rm cm^{-3}$
produces a conventional adiabatic bubble (AB; Figure 5); and the other is a
strongly cooling model, with $V=500\ {\rm\,km\,s^{-1}}$,
$\log\dot{M}=-2~{}/\rm\rm\,M_{\odot}\ yr^{-1}$, and $n=500\ \rm cm^{-3}$ (CB;
Figure 6). The density and temperature profiles for these models are also
shown, and the radial zones corresponding to the freely expanding SSC wind
with the given $\dot{M}$ and $V$, the shock-heated hot bubble, and the dense
outer shell are indicated on the density profile in Figure 5. Figure 6
demonstrates that the CB model’s temperature is below the adiabatic prediction
in the expanding wind region. Thus, its hot bubble temperature is several
times lower, on the order of a few $\times 10^{6}$ K, than that of the AB
model, which has $T>10^{7}$ K. We also see that the CB hot bubble region
occupies a much lower fractional volume interior to the shell. For stronger
cooling, the hot region is cooler and shrinks further.

The emissivity calculations in the bottom panels include both kinetic and
photoionizing activation in the evolving outflow. We see that for the fully
adiabatic model, essentially all the C iv $\lambda 1550$ emission comes from
the shell, and virtually none from the interior (Figure 5). In contrast, for
the strongly cooling model, the interior contributes substantally to the C iv
emission over a large volume (Figure 6; Gray et al., 2019; Danehkar et al.,
2022).

The top panels in Figures 5 and 6 show emissivities of only photoionization
for the same density distribution. This model is largely isothermal at $T\sim
10^{4}$ K, and demonstrates the difference when excluding kinetic heating.
These are useful for inferring the emissivity profiles for pure momentum-
conserving evolution (MC), where there is no shock heating. In that case, the
line emission should generally follow the $r^{-2}$ profiles in the inner
radial zone, which corresponds to the driving wind’s density profile; for
complete catastrophic cooling, i.e., pure MC evolution, this zone would extend
directly to the shell, with no hot bubble zone.

To measure the nebular C iv flux around Knot A, we spatially interpolate over
the stars within the circular emitting region of radius $0.\arcsec 8$ (13 pc),
obtaining a total observed flux of $3.33\pm 0.50\times 10^{-14}\
\rm{\rm\,erg\,s^{-1}}\ cm^{-2}$. We apply the foreground $E(B-V)=0.033$ using
the Milky Way extinction law (Cardelli et al., 1989) and local $E(B-V)=0.084$
for Knot A following Smith et al. (2023), using the SMC reddening from Gordon
et al. (2003). This yields a dereddened flux of $6.84\pm 1.03\times 10^{-14}\
\rm{\rm\,erg\,s^{-1}}\ cm^{-2}$ and surface brightness of $3.41\pm 0.51\times
10^{-14}\ \rm{\rm\,erg\,s^{-1}}\ cm^{-2}\ arcsec^{-2}$. The uncertainties
account for measurement errors only; we estimate that systematic uncertainties
due to continuum subtraction, stellar interpolation, and reddening correction
are on the order of 50%, 10%, and 30%, respectively. We also caution that
there is uncertainty due to the imperfect PSF subtraction described above, but
it is difficult to quantify and Knot A does not have many UV-bright stars in
its immediate vicinity relative to the 0.25$\arcsec$ limit at which the PSF
effect is seen.

Resolved surface brightness can be written as $3.74\times
10^{-12}\epsilon_{i}\ dr$ (Ferland, 2013), where $\epsilon_{i}$ is the C iv
$\lambda 1550$ emissivity and $dr$ is the line-of-sight path length. A value
of $\epsilon_{i}\sim 1\times 10^{-22}\ \rm erg\ s^{-1}\ cm^{-3}$ (see Figure
6) integrated through 20 pc yields a surface brightness $\sim 2\times
10^{-14}\ {\rm\,erg\,s^{-1}}\ \rm cm^{-2}\ arcsec^{-2}$, which agrees well
with the observed values. Given that the predicted values are for the
relatively crude approximations taken from our model grid, this serves to
demonstrate good general consistency between expectations and observations.

Figure 3 shows that the C iv nebular emission around Knot A appears more
consistent with an internally emitting morphology than a limb-brightened,
dense shell. We can also compare with the He ii $\lambda 4686$ morphology,
which, similar to C iv, has only shell emission for strong feedback and
interior emission for weak, radiatively cooling feedback (bottom panels of
Figures 5 and 6, respectively). Figure 7 shows the ratio map of continuum-
subtracted, nebular C iv $\lambda 1550$ / He ii $\lambda 4686$ using the
archive image in F469N from James et al. (2016). These data show that C iv/He
ii is lower in the center around Knot A and higher at larger radii. The
centrally concentrated morphology of He ii was noted by James et al. (2016),
and it is not apparent in C iv (Figure 3). The region with reduced C iv/He ii
extends to a radius $>0.5\arcsec$, which is much larger than the barely
resolved $\sim 0.1\arcsec$ Knot A. The He ii imaging was obtained with the
WFC3 camera, which does not have the PSF wing issue described above for the
ACS SBC MAMA detector, and so the flux cannot be attributed to the He ii-
emitting stars (Smith et al., 2023) in the core.

As seen in Figure 8, this pattern seen in the ratio map is consistent with
that expected for the strongly cooling CB model. The radial morphology of both
C iv and He ii in the CB model is based on the original $r^{-2}$ wind density
profile seen in the upper panels of Figures 5 and 6 for pure photoionization,
which is then modified by kinetic heating such that the central zones become
more highly ionized (Gray et al., 2019), depressing the emissivities for these
species in the center, as seen in the bottom panels of the figures. They also
show enhanced emission at larger radii. However, the two ions have slightly
differing radial profiles, resulting in the radial dependence of their ratio
shown in Figure 8 for the CB model (blue), and which is seen in our data
(Figure 7).

Smith et al. (2023) determine the likely presence of very massive stars (VMS)
from our STIS spectrum. These may have masses up to $600\rm\,M_{\odot}$, and
thus may contribute to the high ionization implied by C iv and He ii emission
(e.g., Berg et al., 2019). In contrast, for the classical AB model, the
emissivity ratio for C iv/He ii is essentially zero for most of the bubble
volume (Figure 8), and the interior emissivities, including in the central
wind region, are orders of magnitude lower and undetectable (Figure 5).

For the CB model in Figure 6, He ii $\lambda 1640$ is brighter than C iv
$\lambda 1550$ at the smallest radii. However, the STIS spectrum (Figure 2)
shows no detection of He ii $\lambda 1640$, even near the SSC itself. This
does not rule out catastrophic cooling, since the central He ii $\lambda 1640$
is not always stronger than C iv $\lambda 1550$ for such models (Danehkar et
al., 2022; Gray et al., 2019). On the other hand, the pure photoionization
models (top panels of Figures 5 and 6) have C iv $\lambda 1550$ / He ii
$\lambda 1640>1$ in the entirety of the central wind zone. This may suggest
that pure photoionization dominates here for Knot A. This is supported by the
fact that O iii] $\lambda\lambda 1661,1666$ is also seen in this region. While
O iii] is seen across the entire Knot A environment, much of this flux is
likely to be foreground and background emission since the entire object is
enveloped in a large halo of more diffuse, extended [O iii] (Figure 1);
however, the central enhancement is coincident with the strongest C iv
emission around the SSC (Figure 2) and suggests local O iii] emission from
Knot A itself. It is important to note that the models are idealized and
remnant high-density gas is known to be in the immediate vicinity of the SSC,
which remains largely enshrouded, at least along the line of sight (e.g.,
Smith et al., 2023; Micheva et al., 2017). Pure photoionization is consistent
with complete cooling of any energy-driven, mechanical feedback.

Figure 7:  Ratio map of continuum-subtracted C iv $\lambda 1550$ / He ii
$\lambda 4686$, with the locations of Knots A and B marked. The central
enhancement around Knot B is an artifact (see Figure 2 and Section 4).  Figure
8:  Predicted emissivity ratio of C iv to He ii as a function of radius for
the AB (red) and CB (blue) models shown in Figures 5 and 6, respectively. Pure
photoionization models are shown in the top panel and combined collisional $+$
photoionization are shown in the bottom panels. We caution that the
emissivities of these lines are orders of magnitude lower in the AB model, as
seen in Figure 5.

Overall, the detection of significant diffuse, nebular C iv emitted from gas
within an SSC-generated cavity must be linked to the absence of pure energy-
driven feedback. In such a scenario, the C iv emission originates either from
catastrophic radiative cooling that disrupts adiabatic feedback, or from pure
photoionization; both scenarios are incompatible with the shock-heated
temperatures ($\gtrsim 10^{6}$ K) required for the classical, adiabatic model,
which would ionize C to higher levels. These conclusions are therefore robust
to parameters like shell geometry and metallicity variations.

The Knot A system has parameters that are quantitatively very similar to those
of the embedded SSC and surrounding molecular gas of NGC 5253 D-1 (e.g.,
Turner et al., 2017). In particular, the molecular velocity dispersion of
$\sim 10\ {\rm\,km\,s^{-1}}$ and gas mass $\sim 10^{5}\ \rm\,M_{\odot}$
observed to be within a few pc of Knot A (Oey et al., 2017) are similar to
values for NGC 5253 D-1, which (Silich et al., 2023) show are consistent with
radiative, catastrophic cooling conditions. The catastrophic cooling models
develop in our grid for $\log\dot{M}=-2\ /\rm\rm\,M_{\odot}\ yr^{-1}$ and
$V\leq 500\ {\rm\,km\,s^{-1}}$. This is quantitatively consistent with the
models for NGC 5253-D1 by Silich et al. (2020), who find that a mass
deposition rate on the order of $\dot{M}\sim 10^{-2}\ \rm\rm\,M_{\odot}\
yr^{-1}$ prevents individual stellar winds from merging and developing a
global outflow. This would also be linked to a strongly cooling environment.

This effective $\dot{M}$ is roughly 2 orders of magnitude higher than what is
expected for a $10^{5}\ \rm\,M_{\odot}$ SSC, and $V$ may be up to an order of
magnitude lower than expected for stellar wind velocities on the order of 2000
${\rm\,km\,s^{-1}}$. These factors are quantitatively consistent with
conservation of the SSC wind’s kinetic energy: a factor of 100 increase in
mass balances a factor of 10 reduction in wind velocity to conserve total
kinetic energy. This suggests that extreme mass-loading of the wind is
directly linked to its deceleration or kinetic inefficiency. The most
straightforward process would be by ablation and shredding of material from
molecular gas clouds and clumps, which are known to be in close proximity to
the SSC and interior to the shell, at least in the line of sight (Oey et al.,
2017). Disks of pre-main sequence stars are also suggested as a source of
material for mass loading (Silich et al., 2020).

However, metal-poor stellar winds may have lower velocities, on the order of
1000 ${\rm\,km\,s^{-1}}$ (e.g., Garcia et al., 2014). Thus, $V$ may be as
little as a factor of 2, instead of an order of magnitude, below that expected
for the SSC. It is therefore likely that mass-loading is enhanced by another
process, in particular, by photoevaporation, which is expected in this
radiation-dominated environment that includes large quantities of molecular
gas. Moreover, radiation is dynamically important, as indicated by the extreme
ionization parameter (Dopita et al., 2002; Yeh & Matzner, 2012) associated
with Knot A, and by the fact that radiation is also believed to concurrently
drive a separate, very low-density, fast superwind through openings in the
shell walls (Komarova et al., 2021). Radiation therefore likely contributes to
maintaining the outward momentum of the slow, mass-loaded flow associated with
the shell around Knot A.

##  4 Knot B and Other Features

Although its mass is up to 10$\times$ lower than Knot A (Micheva et al., 2017;
Drissen et al., 2000), the Knot B SSC is by far the brightest source in Figure
3. The emission from this cluster is dominated by 3 known WR stars identified
by Drissen et al. (2000), including WR3, an unusually luminous ($M_{V}\sim-7$)
WC star. We caution that although many stars are clearly detected in the
continuum-subtracted image, they are not necessarily C iv-emitting stars. As
described in Section 3, the continuum subtraction is calibrated to the
diffuse, nebular C iv emission. This continuum normalization is not optimized
to identify stellar C iv emission, since most such stars have extremely blue
continuum slopes and the long-pass filter bandpasses are very broad. Thus the
point sources in Figure 3 are mostly very FUV-bright stars. Some of the
brightest objects in the figure may be C iv-emitters, but without
spectroscopic confirmation, they should only be considered candidates. These
include the luminous blue variable LBV-V1 (Drissen et al., 1997; Petit et al.,
2006); P-Cygni C iv emission has sometimes been observed in other LBVs, (e.g.,
HD 5980; Koenigsberger et al., 1995). We will carry out detailed analysis of
the stellar population, including its FUV properties, in a follow-up paper.

Figures 1, 3, and 7 appear to suggest the existence of substantial diffuse C
iv emission near the central SSC in Knot B. As noted earlier, this emission is
not seen in the STIS spectrum (Figure 2) and is therefore a spurious effect
likely linked to the large, faint PSF wings of the MAMA detector and the more
complex, extended morphology of this SSC, with numerous FUV-bright stars in
the vicinity of the central core, consistent with its older age. Thus, the
appearance of a high C iv/He ii ratio around Knot B in Figure 7 is an
artifact. The F469N image obtained with WFC3 does not have broad PSF wings and
therefore does not suffer from this effect.

The large, dense shell around Knot B (Figure 1) is likely due to the action of
multiple supernovae from this somewhat older ($\gtrsim 4$ Myr; Micheva et al.,
2017) SSC. This is supported by the existence of diffuse, soft X-ray emission
associated with the southern region of this shell (Thuan et al., 2014), which
is a clear signature of adiabatic, energy-driven feedback.

The lower panels of Figure 5 show the typical relative emissivities for AB
models. While the radial ranges of the different wind and hot bubble zones may
vary depending on the specific parameters, the radial morphologies with the
three shown zones are all similar for strong, adiabatic feedback. In
particular, C iv is ordinarily suppressed near the bubble center due to the
prevalence of C v (e.g., Gray et al., 2019). The lack of central, diffuse C iv
around Knot B is therefore fully consistent with adiabatic feedback.
Furthermore, the presence of central He ii $\lambda 1640$ seen in the STIS
spectrum (Figure 2) is also consistent with the AB model in Figure 5. As noted
earlier, the diffuse O iii] emission in the spectrum is likely foreground or
background emission.

However, there is no apparent detection of limb-brightened C iv from the shell
wall, as expected from the cooling interface with the hot, X-ray emitting gas
(e.g., Chu et al., 1994; Danehkar et al., 2022). Figure 5 shows that for an AB
model, the shell’s predicted emissivity is on the order of $10^{21}\ \rm erg\
s^{-1}\ cm^{-3}$; thus for a $\sim 3$-pc line of sight through the limb, the
surface brightness might be around $3\times 10^{-14}\ \rm erg\ s^{-1}\
cm^{-2}\ arcsec^{-2}$, which is similar to that in the cavity around Knot A.
It may be that the emission is a bit fainter than expected and the
observations are not deep enough to detect the emission.

A non-stellar, dusty “hot spot” that is bright in $K$-band (Drissen et al.,
2000) also appears to be detected in nebular C iv emission (Figure 3). It is
compact, with a FWHM $\lesssim 0.15\arcsec$, or 2.5 pc, and the colors
reported by Drissen et al. (2000) of $J-H=0.4$ and $H-K=1.5$ are consistent
with those of compact H ii regions (e.g., Kastner et al., 2008). The total C
iv flux of the object is $>2.81\pm 0.08\times 10^{-15}\ {\rm\,erg\,s^{-1}}\
\rm cm^{-2}$, taking the extinction correction used for Knot A. The quoted
uncertainty includes only measurement error, while systematic uncertainty due
to continuum subtraction is on the order of 10%. However, this is a
substantial lower limit, since the extinction for such a dense object is
unknown and expected to be much higher than for Knot A. As noted earlier, C iv
emission from H ii regions is unusual and implies the presence of higher-
energy photons. Assuming the object is a compact H ii region, it may host one
or more extremely hot, early O stars, perhaps VMS stars; alternatively, the C
iv emission may be generated in the conductive interface of a very compact
shell enclosing a hot bubble that is strongly pressure-confined (e.g., Silich
et al., 2007; Danehkar et al., 2021). Such a system may also generate C iv by
photoionization from the hot gas (Oskinova & Schaerer, 2022). The location of
this object suggests that its formation may be triggered by the shell
expansion due to Knot B.

##  5 Conclusion

Our successful use of the F150LP and F165LP filters to obtain imaging in C iv
$\lambda 1550$ demonstrates the viability of the ACS SBC channel for obtaining
spatially resolved nebular observations in this critical diagnostic emission
line. Extended diffuse emission is seen in the Mrk 71 system, and also a
highly excited, compact H ii region. FUV-bright stars, including some
candidate stellar C iv sources like WR stars, early O stars, and LBV-V1 are
clearly detected in our observations, although further analysis is needed to
determine the extent to which C iv-emitting stars can be distinguished from
stars with steep FUV continua.

For the Knot A system, provided that the observed diffuse C iv emission is
real, as confirmed by the STIS spectrum, its center-filled spatial
distribution and lack of limb-brightened morphology imply that mechanical
feedback is non-adiabatic, as previously suggested by the surrounding shell
kinematics. Moreover, its morphology and observed flux is consistent with
model expectations for strong radiative cooling. The observed lower C iv/He ii
ratio in the center is also characteristic of such cooling. It may be a system
that is not quite a completely cooled, purely momentum-conserving system,
and/or there may be total cooling and pure photoionization, especially within
Knot A itself. These observations therefore provide direct evidence that Knot
A is likely driving a momentum-conserving shell due to catastrophic cooling
that disrupts adiabatic, energy-driven kinematics.

The Knot B SSC has generated a well-defined superbubble associated with
diffuse X-rays. There is no perceptible C iv emission from within the
superbubble, while central He ii $\lambda 1640$ is detected, as expected from
the SSC wind. These factors all point to a system dominated by adiabatic,
supernova-driven feedback. However, there is no immediate evidence of limb-
brightened C iv emission from the superbubble shell that is predicted for
adiabatic models. Deeper observations are needed to establish the extent to
which this represents a significant discrepancy with predictions.

We thank Will Gray, Genoveva Micheva, and Megan Reiter for useful discussions,
and Roberto Avila of the STScI Astrodrizzle team for advice on the drizzle
procedure. We also thank the anonymous referee for useful comments and
questions. This work was supported by NASA HST-GO-16261. S.S. is supported by
CONAHCYT, México, research grant A1-S-28458. This research is based on
observations made with the NASA/ESA Hubble Space Telescope obtained from the
Space Telescope Science Institute, which is operated by the Association of
Universities for Research in Astronomy, Inc., under NASA contract NAS 5–26555.
These observations are associated with program 16261. The data presented in
this paper were obtained from the Mikulski Archive for Space Telescopes (MAST)
at the Space Telescope Science Institute. The specific observations analyzed
can be accessed via https://doi.org/10.17909/90na-ch15 (catalog DOI:
10.17909/90na-ch15).

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