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# Holonomy groups of compact flat solvmanifolds

A. Tolcachier<EMAIL_ADDRESS>FaMAF, Universidad Nacional de
Córdoba, Ciudad Universitaria, X5000HUA Córdoba, Argentina

###### Abstract.

In this article we study the holonomy group of flat solvmanifolds. It is known
that the holonomy group of a flat solvmanifold is abelian; we give an
elementary proof of this fact and moreover we prove that any finite abelian
group is the holonomy group of a flat solvmanifold. Furthermore, we show that
the minimal dimension of a flat solvmanifold with holonomy group
$\mathbb{Z}_{n}$ coincides with the minimal dimension of a compact flat
manifold with holonomy group $\mathbb{Z}_{n}$. Finally, we give the possible
holonomy groups of flat solvmanifolds in dimensions 3, 4, 5 and 6; exhibiting
in the latter case a general construction to show examples of non cyclic
holonomy groups.

###### 2010 Mathematics Subject Classification:

20H15, 22E25, 22E40, 53C29

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Bieberbach groups and compact flat manifolds
    2. 2.2 Lie groups and flat solvmanifolds
  3. 3 Holonomy of flat solvmanifolds

##  1\. Introduction

A solvmanifold is defined as a compact homogeneous space
$G\backslash\varGamma$ of a simply connected solvable Lie group $G$ by a
discrete subgroup $\varGamma$. Solvmanifolds generalize the well known
nilmanifolds which are defined similarly when $G$ is nilpotent. Both
nilmanifolds and solvmanifolds have provided a large number of examples and
counterexamples in differential geometry . For instance, the first example of
a symplectic manifold without Kähler structure, the so-called “Kodaira-
Thurston manifold”, is a four dimensional nilmanifold [Th]. However, many
important global properties of nilmanifolds cannot be generalized to
solvmanifolds, and for this reason these manifolds are currently widely
studied. One of the most important features of nilmanifolds is a result proved
by Nomizu, which says briefly that the cohomology of the nilmanifolds can be
computed using invariant differential forms, since it is isomorphic to the
cohomology of its Lie algebra [No54]. Unfortunately this is not true in
general for solvmanifolds, but only in particular cases. For example, when $G$
is completely solvable [Hat] or when the “Mostow condition” holds [Mo61].

Solvmanifolds have had several applications in complex geometry. For instance
let us mention the well known Oeljeklaus-Toma manifolds. In [OT], for any
integers $s>0$ and $t>0$, Oeljeklaus and Toma constructed compact complex
manifolds by using number theory and they showed that for $t=1$ these
manifolds admit locally conformally Kähler metrics; they provide a
counterexample to a conjecture made by Vaisman. Kasuya proved in [K] that OT-
manifolds can be represented as solvmanifolds and using this description he
proved that they do not admit any Vaisman metric.

Solvmanifolds have also had applications in theoretical physics, for instance
in [FMOU], solutions of the supersymmetric equations of type II A are
exhibited on six-dimensional solvmanifolds admitting an invariant symplectic
half-flat structure. In [OUV] the authors provide an example of a six-
dimensional complex solvmanifold which admits many invariant solutions to the
Strominger system with respect to certain family of Hermitian connections in
the Green-Schwarz anomaly cancelation condition. Moreover, some of these
solutions satisfy in addition the heterotic equations of motion.

We are interested in flat solvmanifolds, i.e, solvmanifolds admitting a flat
Riemannian metric induced by a flat left invariant metric on the corresponding
Lie group. Milnor [Mi] gave a characterization of Lie groups which admit a
flat left invariant metric and he showed that they are all solvable of a very
restricted form, proving that its Lie algebra decomposes orthogonally as an
abelian subalgebra and an abelian ideal, where the action of the subalgebra on
the ideal is by skew-adjoint endomorphisms. Moreover, such a Lie group $G$ is
of type (I), that is the eigenvalues of $\operatorname{ad}_{x}$ are pure
imaginary for all $x\in\operatorname{Lie}(G)$, and in particular $G$ is
unimodular. Some simply connected Lie groups of this class admit lattices
(i.e. cocompact discrete subgroups), therefore the corresponding solvmanifolds
admit a flat Riemannian metric induced by the flat left invariant metric on
the Lie group.

Flat solvmanifolds are a special class of compact flat manifolds. A classical
theorem asserts that a complete Riemannian manifold is flat if and only if its
universal covering is isometric to Euclidean space. In particular, the
manifold is isometric to a quotient of the form $\mathbb{R}^{n}/\Gamma$ for
some discrete subgroup $\Gamma$ of $\operatorname{Iso}(\mathbb{R}^{n})$, and
its fundamental group is isomorphic to $\Gamma$. In the compact case, such
subgroups are well described by the well known Bieberbach’s theorems ([Bi1,
Bi2, Bi3]) and consequently they are called “Bieberbach groups”. Particularly,
a Bieberbach group $\Gamma$ admits a unique maximal normal abelian subgroup
$\Lambda$ of finite index. Furthermore, the finite group $\Gamma/\Lambda$ can
be identified with the Riemannian holonomy group of the compact flat manifold.
A famous result of Auslander and Kuranishi [AusK, Theorem 3] establishes that
every finite group is the holonomy group of a compact flat manifold.

The fundamental group of a solvmanifold $\varGamma\backslash G$ is isomorphic
to the lattice $\varGamma$ and when the solvmanifold is flat it is isomorphic
to a Bieberbach group. In [Aus], fundamental groups of flat solvmanifolds were
characterized.

The purpose of this article is to study the holonomy group of flat
solvmanifolds. It was proved in [Aus] that these finite holonomy groups are
abelian. In §3, by using Milnor’s result mentioned above, we provide an
elementary proof of this fact (Theorem LABEL:holabeliana). The main result in
this section is the proof of the converse. In order to do this, we focus on a
special class of flat solvmanifolds which are obtained as quotients of almost
abelian Lie groups, since for these Lie groups there exists a criterion to
determine the existence of lattices (see [B]). We show that the holonomy group
of a flat almost abelian solvmanifold is cyclic (Theorem LABEL:holonomia_lcm),
and furthermore for $n=p^{k}$ and $n=2p^{k}$, with $p$ a prime and
$k\in\mathbb{N}$, we give an explicit construction of an almost abelian Lie
group and a lattice such that the quotient solvmanifold has holonomy group
$\mathbb{Z}_{n}$ (Theorem LABEL:holpotprimos). Combining these results we
prove the converse, namely, that every finite abelian group is the holonomy
group of a flat solvmanifold (Theorem LABEL:holabelianofinito).

Due to the result of Auslander and Kuranishi, it is interesting to know, given
a finite group $G$, the minimal dimension of a compact flat manifold with
holonomy group $G$. The minimal dimension was determined for several groups,
for example for $\mathbb{Z}_{p}$ with $p$ prime in [Ch], and more generally
for cyclic groups in [Hi]. Concerning this, we prove in §4 that the minimal
dimension of a flat solvmanifold with holonomy group $\mathbb{Z}_{n}$
coincides with the minimal dimension of a compact flat manifold with holonomy
group $\mathbb{Z}_{n}$, for $n\geq 3$ (Theorem LABEL:mindim).

The last section of the article (§5) is devoted to studying holonomy groups of
low-dimensional flat solvmanifolds. By using a refined version of Milnor’s
result proved by Barberis, Dotti and Fino in [BDF], we give the possible
holonomy groups of flat solvmanifolds in dimensions 3, 4, 5 and 6. Up to
dimension 5 all flat solvmanifolds are quotients of almost abelian Lie groups,
therefore their holonomy groups are cyclic. On the other hand, in dimension 6
we exhibit examples with non-cyclic abelian holonomy groups as particular
cases of a general construction.

The results of this article are from part of my bachelor thesis under the
direction of Adrian Andrada. I would like to thank Roberto Miatello and Juan
Pablo Rossetti for their useful suggestions and comments.

##  2\. Preliminaries

In this short section we outline the basic theory of the topics studied in
this article, namely compact flat manifolds and solvmanifolds equipped with
invariant flat metrics.

###  2.1. Bieberbach groups and compact flat manifolds

It is well known that the group of isometries of $\mathbb{R}^{n}$ is
$\operatorname{Iso}(\mathbb{R}^{n})\cong{\sf
O}(n)\ltimes_{\varphi}\mathbb{R}^{n}$, where $\varphi(A)(v)=Av$. The pair
$(A,v)$ is identified with the isometry $f$ of $\mathbb{R}^{n}$ given by
$f(x)=Ax+v$. This group is called the group of rigid transformations of
$\mathbb{R}^{n}$. Equipped with the product topology of ${\sf O}(n)$ and
$\mathbb{R}^{n}$, $\operatorname{Iso}(\mathbb{R}^{n})$ is a topological group.
The affine group $\mathcal{A}_{n}$ is defined by $\mathcal{A}_{n}={\sf
GL}(n,\mathbb{R})\ltimes_{\varphi}\mathbb{R}^{n}$. An element
$(A,v)\in\operatorname{Iso}(\mathbb{R}^{n})$ is called a pure translation if
$A=\operatorname{Id}$. If $\Gamma$ is a subgroup of
$\operatorname{Iso}(\mathbb{R}^{n})$, the subgroup $\Gamma\cap\mathbb{R}^{n}$
of pure translations of $\Gamma$ will be usually denoted by $\Lambda$, and it
is a torsionfree abelian normal subgroup of $\Gamma$.

######  Definition 2.1.

A subgroup $\Gamma$ of $\operatorname{Iso}(\mathbb{R}^{n})$ is called co-
compact if $\mathbb{R}^{n}/\Gamma$ is compact. A co-compact and discrete
subgroup $\Gamma$ is called a crystallographic group. If $\Gamma$ is also
torsionfree then $\Gamma$ is called a Bieberbach group.

For a discrete subgroup $\Gamma\subset\operatorname{Iso}(\mathbb{R}^{n})$, an
equivalent condition to be torsionfree is that the action of $\Gamma$ on
$\mathbb{R}^{n}$ be free. A nice class of group actions are the properly
discontinuous actions. In general, we say that an action of a subgroup $G$ of
the group of homeomorphisms of a topological space $X$ is properly
discontinuous if for all $x\in X$ there exists an open neighborhood $U$ of $x$
such that $gU\cap U=\emptyset$ for all $g\neq e$. It is known that if a
discrete subgroup $\Gamma$ of $\operatorname{Iso}(\mathbb{R}^{n})$ acts freely
on $\mathbb{R}^{n}$, then the action is properly discontinuous and
$\mathbb{R}^{n}/\Gamma$ admits a structure of a differentiable manifold. As a
corollary, we have the next important theorem about Bieberbach groups.

######  Theorem 2.2.

Let $\Gamma$ be a Bieberbach subgroup of $\operatorname{Iso}(\mathbb{R}^{n})$
and $\pi:\mathbb{R}^{n}\to\mathbb{R}^{n}/\Gamma$ the canonical projection.
Then $\Gamma$ acts on $\mathbb{R}^{n}$ in a properly discontinuous manner,
hence $\pi$ is a covering map and $\pi_{1}(\mathbb{R}^{n}/\Gamma)\cong\Gamma$.
Furthermore $\mathbb{R}^{n}/\Gamma$ admits a structure of a $n$-dimensional
differentiable manifold such that $\pi$ is a local diffeomorphism.

Bieberbach groups, and more generally crystallographic groups, are well
described by three classic theorems known as “Bieberbach’s theorems”. A proof
of them can be found in [Ch].

######  Theorem 2.3 (Bieberbach’s First Theorem).

Let $\Gamma$ be a crystallographic subgroup of
$\operatorname{Iso}(\mathbb{R}^{n})$ and $\Lambda$ the subgroup of pure
translations of $\Gamma$. Then $\Lambda$ is a normal free abelian subgroup of
rank $n$ such that $\Gamma/\Lambda$ is a finite group. Furthermore, $\Lambda$
is maximal as a normal abelian subgroup, in the sense that it contains every
other normal abelian subgroup of $\Gamma$.

The group $\Gamma/\Lambda$ is known as the point group of $\Gamma$ or the
holonomy group of $\Gamma$, in the case that $\Gamma$ is a Bieberbach subgroup
of $\operatorname{Iso}(\mathbb{R}^{n})$. The point group of $\Gamma$ can be
identified with $r(\Gamma)\subset{\sf O}(n)$, which is a finite group, where
$r$ is the projection homomorphism given by $r(A,v)=A.$ Indeed, the kernel of
$r:\Gamma\to{\sf O}(n)$ is $\Lambda$ and thus $\Gamma/\Lambda\cong r(\Gamma)$.

######  Theorem 2.4 (Bieberbach’s Second Theorem).

Let $\Gamma_{1}$ and $\Gamma_{2}$ be crystallographic subgroups of
$\operatorname{Iso}(\mathbb{R}^{n})$ and let $f:\Gamma_{1}\to\Gamma_{2}$ a
group isomorphism. Then there exists $\alpha\in\mathcal{A}_{n}$ such that
$f(\beta)=\alpha\beta\alpha^{-1}$ for all $\beta\in\Gamma_{1}$. In other
words, every isomorphism of crystallographic groups can be realized by an
affine change of coordinates.

######  Theorem 2.5 (Bieberbach’s Third Theorem).

For each $n\in\mathbb{N}$, there are only finitely many isomorphism classes of
crystallographic subgroups of $\operatorname{Iso}(\mathbb{R}^{n})$.

Bieberbach’s theorems are fundamental tools to study compact flat manifolds.
Indeed, a classical theorem asserts that a Riemannian manifold $M$ is flat if
and only if $M$ is isometric to $\mathbb{R}^{n}/\Gamma$ where $\Gamma$ is a
subgroup of $\operatorname{Iso}(\mathbb{R}^{n})$ that acts on $\mathbb{R}^{n}$
in a properly discontinuous manner, that is to say, $\Gamma$ is a Bieberbach
subgroup of $\operatorname{Iso}(\mathbb{R}^{n})$. Furthermore, the holonomy
group of the Riemannian manifold $\mathbb{R}^{n}/\Gamma$ can be identified
with what is called the holonomy group of the Bieberbach group, that is
$\operatorname{Hol}(\mathbb{R}^{n}/\Gamma)\cong r(\Gamma)$, see for instance
[Ch, Example 3.1].

We can reinterpret Bieberbach’s theorems in the context of compact flat
manifolds.

######  Theorem 2.6 (Bieberbach First).

Let $M$ a compact flat manifold. Then $M$ admits a Riemannian covering by a
flat torus and the covering is a local isometry. Furthermore, the holonomy
group $\operatorname{Hol}(M)$ is finite.

######  Theorem 2.7 (Bieberbach Second).

Let $M$ and $N$ be compact flat manifolds with isomorphic fundamental groups.
Then there exists an affine equivalence111An affine equivalence between two
Riemannian manifolds $M$ and $N$ equipped with connections $\nabla^{M}$ and
$\nabla^{N}$ respectively, is a diffeomorphism $F:M\to N$ such that
$\nabla_{X}^{M}Y=\nabla^{N}_{dFX}(dFY)$ for all $X,Y\in\mathfrak{X}(M)$.
between $M$ and $N$ equipped with the corresponding Levi-Civita connections.

######  Theorem 2.8 (Bieberbach Third).

For each $n\in\mathbb{N}$, there are only finitely many classes of affine
equivalence of compact flat manifolds of dimension $n$.

###  2.2. Lie groups and flat solvmanifolds

In the context of Lie groups, the most interesting metrics are the left
invariant ones, which satisfy that the left translations are isometries. When
$G$ is an abelian Lie group, it is clear that every left invariant metric is
flat. An interesting fact proved by Milnor is that some left invariant metrics
may be flat, even though $G$ is not abelian. Indeed, he described the
structure of a Lie group admitting one of these flat metrics (see [Mi, Theorem
1.5]). The precise criterion is stated as follows.

######  Theorem 2.9.

A left invariant metric on a Lie group $G$ is flat if and only if the
associated Lie algebra $\mathfrak{g}$ splits as an orthogonal direct sum
$\mathfrak{g}=\mathfrak{b}\oplus\mathfrak{u}$, where $\mathfrak{b}$ is an
abelian subalgebra, $\mathfrak{u}$ is an abelian ideal and the linear
transformation $\operatorname{ad}_{b}$ is skew-adjoint for every
$b\in\mathfrak{b}$.

We will call $(G,\langle\cdot,\cdot\rangle)$ a flat Lie group if
$\langle\cdot,\cdot\rangle$ is a flat left invariant metric on the Lie group
$G$ and $(\mathfrak{g},\langle\cdot,\cdot\rangle_{e})$ will be called a flat
Lie algebra.

Using Milnor’s characterization of flat Lie algebras, Barberis, Dotti and Fino
decompose even more a flat Lie algebra [BDF, Proposition 2.1].

######  Theorem 2.10.

Let $(\mathfrak{g},\langle\cdot,\cdot\rangle_{e})$ be a flat Lie algebra. Then
$\mathfrak{g}$ splits as an orthogonal direct sum,

$\mathfrak{g}=\mathfrak{z}(\mathfrak{g})\oplus\mathfrak{b}\oplus[\mathfrak{g},\mathfrak{g}]$

where $\mathfrak{b}$ is an abelian subalgebra, $[\mathfrak{g},\mathfrak{g}]$
is abelian and the following conditions are satisfied:

  1. (1)

$\operatorname{ad}:\mathfrak{b}\to\mathfrak{so}[\mathfrak{g},\mathfrak{g}]$ is
injective, $\dim[\mathfrak{g},\mathfrak{g}]$ is even and
$\dim\mathfrak{b}\leq\frac{\dim[\mathfrak{g},\mathfrak{g}]}{2}$;

  2. (2)

$\nabla_{X}=\operatorname{ad}_{X}$ for all $X\in\mathfrak{b}$ and
$\nabla_{X}=0$ for all
$X\in\mathfrak{z}(\mathfrak{g})\oplus[\mathfrak{g},\mathfrak{g}]$.

As a consequence, since $\\{\operatorname{ad}_{X}\mid X\in\mathfrak{b}\\}$ is
an abelian subalgebra of $\mathfrak{so}[\mathfrak{g},\mathfrak{g}]$, it is
included in a maximal abelian subalgebra and as all of these are conjugated,
there exists an orthonormal basis $\mathcal{B}$ of
$[\mathfrak{g},\mathfrak{g}]$ such that for $X\in\mathfrak{b}$,

$[\operatorname{ad}_{X}]_{\mathcal{B}}=\begin{pmatrix}0&a_{1}&&&\\\
-a_{1}&0&&&\\\ &&\ddots&&\\\ &&&0&a_{m}\\\
&&&-a_{m}&0\end{pmatrix},\quad\text{with}\;a_{1},\,\ldots\,,a_{m}\in\mathbb{R},\;m=\frac{\dim[\mathfrak{g},\mathfrak{g}]}{2}.$

Note that a flat Lie algebra $(\mathfrak{g},\langle\cdot,\cdot\rangle_{e})$ is
$2$-step solvable, since $[\mathfrak{g},\mathfrak{g}]$ is abelian; and
unimodular222A Lie algebra $\mathfrak{g}$ is said to be unimodular if
$\operatorname{tr}\operatorname{ad}_{X}=0$ for all $X\in\mathfrak{g}$., since
$\operatorname{ad}_{X}$ is skew-adjoint for all $X\in\mathfrak{g}$.

Now, we will see that we can obtain a special class of compact flat manifolds
as quotients of Lie groups with a flat invariant metric. We recall first a
theorem about the differentiable structure of a quotient manifold
$G/\varGamma$ in the case that $\varGamma$ is a closed subgroup of the Lie
group $G$.

######  Theorem 2.11.

Let $\varGamma$ be a closed subgroup of a Lie group $G$. The left coset space
$G/\varGamma$ is a topological manifold of dimension equal to $\dim
G-\dim\varGamma$, and has a unique smooth structure such that the quotient map
$\pi:G\to G/\varGamma$ is a smooth submersion.

When $\varGamma$ is a discrete subgroup of a Lie group $G$, hence closed and
$0$-dimensional, the quotient map $\pi:G\to G/\varGamma$ is a local
diffeomorphism. Let us assume now that $G$ carries a left invariant Riemannian
metric. Instead of working with the left coset space, we will consider the
right coset space $\varGamma\backslash G$ (note that the theorem above is also
valid for the right coset space), to be able to induce a Riemannian metric on
$\varGamma\backslash G$ so that $\pi$ results an isometry: Given $g\in G$ and
$u,v\in T_{\pi(g)}(\varGamma\backslash G)$, we define

$\langle
u,v\rangle_{\pi(g)}=\langle(d\pi)_{g}^{-1}u,(d\pi)_{g}^{-1}v\rangle_{g}.$

The metric is well defined since $L_{\gamma}$ is an isometry of $G$ and
$\pi\circ L_{\gamma}=\pi$ for all $\gamma\in\varGamma$, where
$L_{\gamma}(g)=\gamma g$ for all $g\in G$. It is clear that $\pi$ is a local
isometry. In particular, if the metric on $G$ is flat, the induced metric on
$\varGamma\backslash G$.

The most interesting case is when $\varGamma$ is a co-compact discrete
subgroup of $G$, which is called a lattice of $G$.

######  Definition 2.12.

A solvmanifold $M$ is a compact quotient $\varGamma\backslash G$ where $G$ is
a simply connected solvable Lie group and $\varGamma$ is a lattice of $G$.
When $G$ is nilpotent, $M$ is said to be a nilmanifold.

######  Remark 2.13.

It is worth mentioning that in the literature, different definitions of
solvmanifolds are also considered. For example, L. Auslander defines a
solvmanifold as a “homogeneous space of connected solvable Lie group”, [Aus61,
p. 398], see also [Aus73a, Aus73b]. We will only consider solvmanifolds as in
our definition above, and as a consequence they are always compact, orientable
and paralellizable.

Every simply connected solvable Lie group is diffeomorphic to $\mathbb{R}^{n}$
(see for instance [Va]) for some $n\in\mathbb{N}$. This implies that a
solvmanifold is aspherical, that is, the higher homotopy groups vanish; and
$\pi_{1}(\varGamma\backslash G)\cong\varGamma$. The fundamental group plays an
important role in the study of solvmanifolds. Indeed, Mostow’s Theorem show
that the solvmanifolds are classified, up to diffeomorphism, by their
fundamental groups. A proof of this can be found in [Rag, Theorem 3.6].

######  Theorem 2.14 (Mostow).

Let $G_{1}$ and $G_{2}$ be simply connected solvable Lie groups with
$\varGamma_{i}$ a lattice in $G_{i}$ for $i=1,2$. If
$\phi:\varGamma_{1}\to\varGamma_{2}$ is an isomorphism then there exists a
diffeomorphism $\tilde{\phi}:G_{1}\to G_{2}$ such that
$\tilde{\phi}|_{\varGamma_{1}}=\phi$ and $\tilde{\phi}(\gamma
g)=\phi(\gamma)\tilde{\phi}(g)$ for all $\gamma\in\varGamma_{1},g\in G_{1}$.

######  Corollary 2.15.

Two solvmanifolds with isomorphic fundamental groups are diffeomorphic.

Concerning the construction of solvmanifolds and nilmanifolds, Malcev’s
theorem [Ma] asserts that a simply connected nilpotent Lie group $G$ admits a
lattice if and only if its Lie algebra has a basis such that the structure
constants are rational numbers, so it is not so difficult to construct
nilmanifolds. Unfortunately, it is not easy to construct solvmanifolds, since
there is not a general method to find lattices in a given solvable Lie group.
It is known ([Mi]) that a necessary condition for a solvable Lie group to
admit lattices is to be unimodular, but in general there are no known
sufficient conditions.

A special class of solvmanifolds that we are going to consider later are those
obtained as quotients of almost abelian Lie groups, which we will call almost
abelian solvmanifolds.

######  Definition 2.16.

A non-abelian Lie algebra $\mathfrak{g}$ is said to be almost abelian if
$\mathfrak{g}$ has a codimension 1 abelian ideal, that is
$\mathfrak{g}=\mathbb{R}x\ltimes_{\operatorname{ad}_{x}}\mathbb{R}^{n}$ for
some $n\in\mathbb{N}$.

It is clear that an almost abelian Lie algebra is solvable and it is nilpotent
if and only if $\operatorname{ad}_{x}|_{\mathbb{R}^{n}}$ is nilpotent.
Moreover, if we denote by $\mathfrak{g}_{A}$ the almost abelian Lie algebra
with $\operatorname{ad}_{x}|_{\mathbb{R}^{n}}=A$, it can be proved that
$\mathfrak{g}_{A}\cong\mathfrak{g}_{B}$ if and only if there exist
$0\neq\lambda\in\mathbb{R}$ and $C\in{\sf GL}(n,\mathbb{R})$ such that
$B=\lambda CAC^{-1}$.

The simply connected Lie group $G$ corresponding to an almost abelian Lie
algebra
$\mathfrak{g}=\mathbb{R}x\ltimes_{\operatorname{ad}_{x}}\mathbb{R}^{n}$, is
called an almost abelian Lie group, and can be written as
$G=\mathbb{R}\ltimes_{\varphi}\mathbb{R}^{n}$, where
$\varphi(t)=\exp(t\operatorname{ad}_{x})$.

A great advantage of almost abelian Lie groups is that there exists a
criterion to determine whether they admit lattices or not (see [B]).

######  Proposition 2.17.

Let $G=\mathbb{R}\ltimes_{\varphi}\mathbb{R}^{m}$ be an almost abelian Lie
group. Then $G$ admits a lattice $\varGamma$ if and only if there exists
$t_{0}\neq 0$ such that $\varphi(t_{0})$ is conjugate to an integer matrix
invertible in $\mathbb{Z}$. In this case, the lattice $\varGamma$ is given by

$\varGamma=t_{0}\mathbb{Z}\ltimes_{\varphi}P\mathbb{Z}^{m},$

where $P^{-1}{\varphi(t_{0})}P$ is an integer matrix.

##  3\. Holonomy of flat solvmanifolds

In this section we are going to consider solvmanifolds $\varGamma\backslash G$
equipped with a flat metric induced by a flat left invariant metric on $G$,
using the characterization given by Milnor mentioned above.

We do not assume that $G$ acts by isometries on $\varGamma\backslash G$. In
fact, it is well known that if $M$ is a flat Riemannian manifold with a
transitive group of isometries, then $M\cong T^{m}\times\mathbb{R}^{k}$ for
some $k,m\in\mathbb{N}$ (see [Wo]).

In general, if a simply connected Lie group $G$ admits a flat left invariant
metric, then $G$ is isometric as a Riemannian manifold to $\mathbb{R}^{n}$
with the usual metric. We can identify $\varGamma$ with a subgroup of
isometries of $\mathbb{R}^{n}$ as follows:

Let $\psi:G\to\mathbb{R}^{n}$ an isometry. Define a map
$i:G\to\operatorname{Iso}(\mathbb{R}^{n})$ by $i(g)=\tau_{g}$, where
$\tau_{g}=\psi\circ L_{g}\circ\psi^{-1}$. It is clear that $i$ is an injective
group homomorphism, and therefore we can identify $\varGamma$ with its image
and consider $\varGamma\subset\operatorname{Iso}(\mathbb{R}^{n})$.

The solvmanifold $\varGamma\backslash G$ is isometric to
$\mathbb{R}^{n}/\varGamma$ by considering the map $\phi([g])=[\psi(g)]$.
Observe that $\phi$ is well defined since

$\phi([\gamma g])=[\psi(\gamma
g)]=[\tau_{\gamma}(\psi(g))]=[\psi(g)]=\phi([g]).$

Furthermore it is injective, since if $[\psi(g)]=[\psi(h)]$ then
$\psi(g)=\tau_{\gamma}(\psi(h))$, which implies $\psi(g)=\psi(\gamma h)$.
Thus, $g=\gamma h$, or equivalently, $[g]=[h]$. The surjectivity is clear, and
as $\phi$ is a composition of local isometries and bijective, $\phi$ results
an isometry. Therefore $\varGamma$ is a Bieberbach subgroup of
$\operatorname{Iso}(\mathbb{R}^{n})$ and
$\operatorname{Hol}(\mathbb{R}^{n}/\varGamma)\cong r(\varGamma)$. We recall
that $r(\varGamma)$ is identified with the finite group $\varGamma/\Lambda$
where $\Lambda$ is the maximal normal abelian subgroup of $\varGamma$. In
conclusion,

(1) $\operatorname{Hol}(\varGamma\backslash G)\cong\varGamma/\Lambda.$


# Probability Series Expansion Classifier that is Interpretable by Design

Sapan Agarwal
Sandia National Laboratories
PO Box 969 MS 9158
Livermore, CA 94551-0969
<EMAIL_ADDRESS>
&Corey M. Hudson
Sandia National Laboratories
PO Box 969 MS 9291
Livermore, CA 94551-0969
<EMAIL_ADDRESS>

###### Abstract

This work presents a new classifier that is specifically designed to be fully
interpretable. This technique determines the probability of a class outcome,
based directly on probability assignments measured from the training data. The
accuracy of the predicted probability can be improved by measuring more
probability estimates from the training data to create a series expansion that
refines the predicted probability. We use this work to classify four standard
datasets and achieve accuracies comparable to that of Random Forests. Because
this technique is interpretable by design, it is capable of determining the
combinations of features that contribute to a particular classification
probability for individual cases as well as the weightings of each of
combination of features.

##  1 Introduction

When exploring the decisions a machine-learning classifier makes, it is often
challenging to understand the individual choices that contribute to the
decision, for all but the most trivial classifiers [1-3]. This is increasingly
problematic as machine-learning classifiers handle more critical [4] and
socially important systems [5], and as governments [6] and citizens demand
more oversight [7]. The problem emerges because most classifiers are trained
through the solution of a complex optimization problem that is designed to fit
a large number of model parameters. In order to avoid over-fitting the data,
the data is regularized and learning rules are extracted, thereby reducing the
overall information content in hopes of achieving relevant, rather than exact
models [8]. This process inherently obscures the properties of the training
data that leads to the classification, resulting in the need for heuristic
methods to try to interpret the classifier. Simple decision tree based
classifiers have a straight-forward interpretation, but in order to get high
classification accuracies and reduce over-fitting, the results of many trees
need to be combined, thus obscuring any attempts to interpret the model.

To overcome these limitations, we propose a new classifier that estimates the
probability of a classification outcome based directly on probabilities
measured from the training data. We will show that this classifier has an
accuracy performance that is comparable to random forests. Additionally, this
classifier allows us to determine the combinations of features that
contributed to the particular classification as well as their weights.

##  2 Series Classifier Model

The goal of the new classifier is to predict an outcome $Y=Y^{\prime}$, given
measured categorical features
$x_{1}=x_{1}^{\prime},\>x_{2}=x_{2}^{\prime},\>x_{3}=x_{3}^{\prime}$,… In
particular, we want to estimate:

$P(Y=Y^{\prime}|x_{1}=x_{1}^{\prime},x_{2}=x_{2}^{\prime},x_{3}=x_{3}^{\prime},\ldots)$
(1)

using only probabilities measured from the training data. We seek to measure
the probability that $Y=Y^{\prime}$ given different combinations of features
such as: $P(Y=Y^{\prime})$, $P(Y=Y^{\prime}|x_{1}=x_{1}^{\prime})$,
$P(Y=Y^{\prime}|x_{3}=x_{3}^{\prime})$,
$P(Y=Y^{\prime}|x_{1}=x_{1}^{\prime},\>x_{2}=x_{2}^{\prime})$, … The key
challenge is deciding how to combine these measured probabilities into a
singular estimate. Unfortunately, there are no good first principles methods
for combining these probabilities. Trying to use Bayesian methods to put
limits on Eq. (1) will often give limits of 0 and 1, giving no useful
information. The simplest option is to average probabilities:

$P(Y=Y^{\prime}|x_{1}=x_{1}^{\prime},x_{2}=x_{2}^{\prime},x_{3}=x_{3}^{\prime},\ldots)\approx\left.{\sum\nolimits_{i}P(Y=Y^{\prime}|x_{i}=x_{i}^{\prime})}\middle/{\sum\nolimits_{i}1}\right.$
(2)

However, this will not account for a number of effects as described in the
following sections. To account for those effects, we will hierarchically
weight the averages of the contributing probabilities. First, we define the
following notation:

and so on. Furthermore, we define $P^{\prime}$ as the estimate of $P$ after
accounting for $P_{i}^{\prime}$, $P_{i}^{\prime}$ is the estimate of $P_{i}$
after accounting for $P_{ij}^{\prime}$ and so on.

###  2.1 Create a Series Expansion of Probabilities

To account for multiple levels of measured probabilities, the probability
estimate can hierarchically be refined as follows:

The overall probability estimate is evaluated, starting at the lowest level
($P(Y=Y^{\prime})$ given the largest number of features) for which there is a
measured probability and working upwards until the estimate $P^{\prime}$ is
computed. The sums are simply taken over probabilities that have been measured
in the training data. If an estimated probability such as $P_{ij}^{\prime}$,
is not known, the directly measured value ${P_{ij}}$ should be used where
available.

If all possible combinations of features are kept, the size of the model would
grow exponentially as the number of levels in the model increases. To limit
the model size, we selectively choose the probabilities to keep track of,
based on which combinations of features minimize Gini Impurity (or entropy).
In the tests performed later, we kept all possible probabilities, $P_{i}$ and
$P_{ij}$ for the first two levels and then kept only the top six $P_{ijk}$ for
each $P_{ij}$ and the top six $P_{ijkl}$ for each $P_{ijk}$ used. Missing
features can easily be handled by leaving out probabilities corresponding to
the missing features.

###  2.2 Weight P=0 and P=1 More Strongly

If any feature defines $P=0$ or $P=1$, then the overall probability is 0 or 1,
regardless of other features. Consequently, when averaging probabilities in
Eq. (4), the closer a probability is to 1 or 0, the more it should be
weighted. Thus, we introduce the following scaling weight $S_{i}$:

$S_{i}=\left.{1}\middle/[{P_{i}^{\prime}\times(1-P_{i}^{\prime})}]\right.$ (5)

The probability computation Eq. (4a) is updated as follows:

$P^{\prime}={\sum_{i}{P_{i}^{\prime}\times
S_{i}}}\bigg{/}{\sum\nolimits_{i}{S_{i}}}$ (6)

A logarithmic scaling that increases more slowly as $P$ approaches 0 or 1 was
included as a heuristic in earlier versions of the algorithm, but was found to
lower classifier performance.

###  2.3 Account for Noise and Low Data Counts

Each probability estimate, $P_{i}$, $P_{ij}$, etc. is backed by some number of
training examples, $n_{train}$. If this number is too low, noise can result in
a poor probability estimate. It is also possible to get $P=0$ or $P=1$ for
some combinations of features, which could cause the scaling factor Eq. (5) to
blow up. To account for this we need to make three corrections: 1) Eq. (5)
needs to be updated to account for the number of training examples 2) Each
probability estimate should be weighted by the amount of data backing it and
3) If there is not enough data backing a probability estimate its parent
estimate should be used (i.e. use $P_{i}$ instead of $P_{ij}$)

First, we need to limit the $P_{i}^{\prime}$ used in Eq. (5) based on the
number of training samples backing $P_{i}$:

$S_{i}=\left.{1}\middle/[P_{i}^{{}^{\prime\prime}}\times(1-P_{i}^{{}^{\prime\prime}})]\right.$
(7a) $P_{i}^{{}^{\prime\prime}}=\begin{dcases}0.5&n_{train,i}\leq n_{err}\\\
{n_{err}}/{N_{train,i}}&P_{i}^{\prime}<{n_{err}}/{N_{train,i}}\text{ and
}n_{train,i}>n_{err}\\\ P_{i}^{\prime}&{n_{err}}/{N_{train,i}}\leq
P_{i}^{\prime}\leq 1-{n_{err}}/{N_{train,i}}\text{ and }n_{train,i}>n_{err}\\\
1-{n_{err}}/{N_{train,i}}&P_{i}^{\prime}>1-{n_{err}}/{N_{train,i}}\text{ and
}n_{train,i}>n_{err}\end{dcases}$ (7b)

Here $n_{err}$ is a is a meta-parameter that represents average number of
examples that might be wrong due to noise. Next, we need to add a weight to
the averages in Eq. (3) based on the amount of data used:

$P^{\prime}=\frac{\sum_{i}P_{i}^{\prime}\times S_{i}\times
N_{i}}{\sum_{i}S_{i}\times N_{i}}$ (8a)
$N_{i}=\min\left(\sqrt{n_{train,i}},\sqrt{n_{max}}\right)$ (8b)

Here $n_{max}$ is a meta-parameter that represents the number of training
examples needed to overcome noise. Finally, if none of the probabilities are
backed by significant data, the parent probability should be used:

$P^{\prime}=\frac{\max\limits_{i}(N_{i})}{N}\times\frac{\sum_{i}P_{i}^{\prime}\times
S_{i}\times N_{i}}{\sum_{i}S_{i}\times
N_{i}}+\left(1-\frac{\max\limits_{i}(N_{i})}{N}\right)\times P$ (9)

###  2.4 Account for the Usefulness of a Feature Combination

If a particular combination of features significantly changes the probability
estimate (i.e. $P_{ij}^{\prime}>>P_{i}$) it can contain new information and
should therefore be weighted more heavily. Consequently, we can optionally add
a usefulness weight, $U_{i}$:

$U_{i}=(P_{i}^{\prime}-P)\>/\>(P_{i}^{\prime}+P)$ (10)

Using this weight was found to improve accuracy for some datasets, but not
all.

Table 1: The new series classifier achieves accuracies comparable to Random Forests Dataset | Series Classifier | Random Forests
---|---|---
1984 Congressional Voting Records [11,12] | 96.8% | 96.8%
E. coli Promotor Gene Sequences [12,13] | 95.3% | 94.3%
SPECT Heart Data [12,14] | 85.0% | 85.0%
Lymphography [12,15] | 86.4% | 87.2%

###  2.5 Overall Model

The overall model accounting for all the above factors is given by:

$\displaystyle P^{\prime}$
$\displaystyle=\frac{\max_{i}(N_{i})}{N}\times\frac{\sum_{i}P_{i}^{\prime}\times
S_{i}\times N_{i}\times U_{i}}{\sum_{i}S_{i}\times N_{i}\times
U_{i}}+\left(1-\frac{\max_{i}(N_{i})}{N}\right)\times P$ (11a) $\displaystyle
P_{i}^{\prime}$
$\displaystyle=\frac{\max_{j}(N_{ij})}{N_{i}}\times\frac{\sum_{j\neq
i}P_{ij}^{\prime}\times S_{ij}\times N_{ij}\times U_{ij}}{\sum_{j\neq
i}S_{ij}\times N_{ij}\times
U_{ij}}+\left(1-\frac{\max_{j}(N_{ij})}{N_{i}}\right)\times P_{i}$ (11b)
$\displaystyle P_{ij}^{\prime}$
$\displaystyle=\frac{\max_{k}(N_{ijk})}{N_{ij}}\times\frac{\sum_{k\neq j\neq
i}P_{ijk}^{\prime}\times S_{ijk}\times N_{ijk}\times U_{ijk}}{\sum_{k\neq
j\neq i}S_{ijk}\times N_{ijk}\times
U_{ijk}}+\left(1-\frac{\max_{k}(N_{ijk})}{N_{ij}}\right)\times P_{ij}$ (11c)

##  3 Classification Results

To benchmark the performance of this new classifier, we compared
classification accuracy against scikit-learn’s [9] Random Forests
implementation [10] on four different datasets (summarized in Table 1). As
seen in the table the new classifier achieves accuracies comparable to Random
Forests. The hyper parameters ($n_{err}$, $n_{max}$, tree depth, and use/don’t
use usefulness model) were optimized using 4-fold cross-validation and the
best accuracies are reported.

##  4 Interpretability Results

Because this classifier assesses combinations of probabilities, it can
directly calculate the probability contributed to the overall classification
estimate and use this to understand which features contributed most highly to
its estimate. To demonstrate this, we analyze an example in the congressional
voting records dataset that was misclassified in Table 2. The classifier
misclassified with 70% probability that this Member of Congress would be a
Republican when they are a Democrat. Table 2 shows the top six feature
combinations that contributed to the classification and the corresponding
weight assigned to each feature. The particular votes that the features
correspond to are given in the Supplementary Information. This shows that in
the training data certain feature combinations indicated with 100% probability
that the Member of Congress was a Republican and other feature combinations
indicated that the Member of Congress was a Democrat with 100% probability. To
understand the impact of each individual feature, given the feature
combinations, the probability estimates, $P_{i}^{\prime}$ and the
corresponding weights are shown in Table 3. These features indicated that the
Member of Congress was a Republican and were weighted more highly resulting in
the classification of this member as a Republican.

Table 2: Analysis of a misclassified instance by feature combination Features |  Counts |  Counts |  Probability |  Weight |  Cumulative
---|---|---|---|---|---
|  Republican |  Democrat |  Republican |  |  Weight
$x_{3}=1$, $x_{9}=1$, $x_{15}=0$ |  16 |  0 |  100% |  12.5% |  12.5%
$x_{9}=1$, $x_{15}=0$ |  16 |  0 |  100% |  7.0% |  19.4%
$x_{3}=1$, $x_{8}=0$, $x_{9}=1$, $x_{14}=0$ |  47 |  0 |  100% |  4.8% |  24.2%
$x_{3}=1$, $x_{7}=0$, $x_{9}=1$, $x_{14}=0$ |  47 |  0 |  100% |  4.7% |  28.9%
$x_{2}=1$, $x_{10}=1$, $x_{11}=0$ |  0 |  58 |  0% |  4.0% |  32.9%
$x_{1}=1$, $x_{2}=1$, $x_{10}=1$ |  0 |  38 |  0% |  3.1% |  36.0%
Table 3: Analysis of a misclassified instance by individual feature Feature # | Probability Republican | Weight | Feature # | Probability Republican | Weight
---|---|---|---|---|---
$x_{9}=1$ | 95% | 17% | $x_{8}=0$ | 86% | 4%
$x_{3}=1$ | 94% | 16% | $x_{7}=0$ | 83% | 4%
$x_{15}=0$ | 94% | 14% | $x_{6}=0$ | 29% | 3%
$x_{2}=1$ | 14% | 8% | $x_{0}=0$ | 78% | 3%
$x_{10}=1$ | 23% | 8% | $x_{13}=1$ | 55% | 2%
$x_{11}=0$ | 21% | 6% | $x_{4}=1$ | 69% | 2%
$x_{14}=0$ | 89% | 5% | $x_{12}=1$ | 52% | 2%
$x_{1}=1$ | 53% | 4% | $x_{5}=1$ | 56% | 2%

##  5 Conclusion

Specifically designing a classifier for interpretability naturally allows for
more highly interpretable results. We show that is possible to design a
classifier based on averaging measured probabilities that gives
classifications which are as accurate as those in Random Forests. Using this
classifier, we can easily determine which features and combinations of
features contribute to a classification and audit and assess the limitations
and strength of the model, identify potential problems in the training data
and explain outlying instances.

#### Acknowledgments

Sandia National Laboratories is a multimission laboratory managed and operated
by National Technology and Engineering Solutions of Sandia, LLC., a wholly
owned subsidiary of Honeywell International, Inc., for the U.S. Department of
Energy’s National Nuclear Security Administration under contract DE-NA0003525.

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## Supplementary Information

Each of the features listed in Tables 2 and 3 correspond to votes on the
following bills. A vote of yes corresponds to a feature value of 1 and a vote
of no corresponds to a feature value of 0.

Table 4: Bill descriptions for each feature number Feature Number | House Bill Description
---|---
0 | Handicapped infants
1 | Water project cost sharing
2 | Adoption of the budget resolution
3 | Physician fee freeze
4 | El Salvador aid
5 | Religious groups in schools
6 | Anti-satellite test ban
7 | Aid to Nicaraguan contras
8 | Mx missile
9 | Immigration
10 | Synfuels corporation cutback
11 | Education spending
12 | Superfund right to sue
13 | Crime
14 | Duty free exports
15 | Export administration act South Africa