Rigidity of actions on metric spaces close to three dimensional manifolds

The question on the nature of group homomorphisms $\rho:\Gamma\to{\rm Diff}(M)$, between a finitely generated group and the group of diffeomorphisms of a compact, $n$-dimensional, smooth manifold is interesting in many contexts. Particularly, in a series of conjectures known as the Zimmer program [Zim87], [Fis11], [Fis20], concerning the question on whether the group homomorphism cannot have large image if the dimension of the manifold is small, relative to the rank of the group. More precisely, the Zimmer program deals with groups $\Gamma$ which are lattices in a semisimple algebraic Lie group of rank at least 2, which we will refer to in this introduction as higher rank lattices see 3. As an example of this, in the recent result [BFH20], it is found that a homomorphism $\rho:SL_{k+1}(\mathbb{Z})\rightarrow{\rm Diff(M)}$ must have finite image when $k>n$ and $k\geq 2$, in this case the parameter $k$ is the rank of the Lie group $SL_{k+1}(\mathbb{R})$. This result is greatly generalized for other higher rank semisimple lattices on [BFH21]. Another instance of the Zimmer program is the complete characterization of the action in critical dimensions, c.f. Conjectures 1.2 and 1.3 in [Fis11]. For example, when the dimension hypothesis is modified in the previous context to $n=k+1$, and the hypothesis that the action preserves a finite volume and an affine connection is added, then [Fis11, Theorem 6.9] tells us that the action is conjugated to the standard linear action of $SL_{n}(\mathbb{Z})$ on $\mathbb{R}^{n}/\mathbb{Z}^{n}$.

The $C^{0}$-version of the Zimmer Program, as suggested in [Wei11], and [Ye20], [Ye19], asks roughly for changing the category of manifolds and morphisms in the Zimmer Program, from the smooth setting into a topological setting, that is, by considering a group homomorphism from a finitely generated group, and specifically a higher rank lattice, onto the group of homeomorphisms within a prescribed category (topological, smooth, piecewise linear, etc.). The following Conjecture is an example of a problem stated in this setting, found in [Ye20]:

This variation of Zimmer Program is, as expected, much harder than the original one, and it has shown only small advances, such as the solution in the one-dimensional case by Hurtado and Deroin [DH20], and in the context of infinite homological actions on three-manifolds [FS00].

It is natural to explore analogous rigidity results outside of the differentiable category and into the category of metric spaces endowed with extra structure. As an example of this, in [Hae20] it is proved that any action of a higher rank lattice in a Gromov $\delta$-hyperbolic metric space must be elementary. Among many things, such result implies that any homomorphism $\rho:\Gamma\rightarrow\textrm{Mod}(S)$ from a higher rank lattice onto the mapping class group of a compact surface is finite, a result first proved in [FM98].

The notion closer to manifolds for which we explore these rifidity results is that of Alexandrov spaces, which are metric spaces with a synthetic notion of curvature bounded from below. Alexandrov spaces include compact Riemannian manifolds and non-trivial modifications of them, such as orbifold quotients and Gromov-Hausdorff limits (see Section 3.2. In this paper we propose the following variation of Problem 2.1

Of course, Problem 2.2 can be stated for general higher rank lattices with comparison bounds on the dimension and rank as in [Fis11, Conjecture 4.12].

Alexandrov spaces are rigid (in a sense analogous to Gromov’s rigid geometric structures [Fis11]) as their isometry groups are Lie groups with bounded dimension in terms of the dimension of the space [GGG13]. Moreover, 3-dimensional Alexandrov spaces which are sufficiently collapsed (c.f. section Alex), are in fact orbifolds over one of the eight Thurston geometries [GGGNnZ20]. These reduction phenomenae together with classification results for isometric actions on Thurston geometries, lead us to the following much more tractable problem:

It is worth mentioning other rigidity results obtained for $3$-dimensional Alexandrov spaces such as the proof of the Borel conjeture for sufficiently collapsed Alexandrov spaces [BNnZ21]. Finally, John Pardon’s proof of the Hilbert-Smith conjecture for three manifolds [Par13], can be extended to the singular case in the setting of Alexandrov spaces as it can be reduced to a local behaviour, leading to the following result:

This result lead to the natural generalization of the Hilbert-Smith conjecture, which simply would ask if Theorem 2.4 is valid for $n$-dimensional Alexandrov spaces). A first approach to this generalizationis to extend the result of [RS97], proving Hilbert-Smith conjecture for Lipschitz actions, where the difficulty lies on the extension of Yang’s Theorem (on the increase of dimension in the quotient for $p$-adic actions [Yan60] [BRW61]) to Alexandrov spaces.

The main result of this paper, concerning Problem 2.3 is

Recall that an isotypic group of type $SO(3)$ is an algebraic group which is, up to finite covers and connected components, a product of copies of $SO(3)$ and $SO(2,1)$ (see Section 3). In [BNnZ21], a three dimensional Alexandrov space $X$ of ${\rm curv}\geq-1$ is said to be sufficiently collapsed, if there exist $D>0$ and $\epsilon>0$ such that the diameter of $X$ is less or equal to $D$, and the volume is strictly less than $\epsilon$. We include as a corollary of the results here a classification of the discrete groups acting by isometries on a three dimensional Alexandrov space with a sufficiently collapsed quotient

As an immediate consequence of this theorem, we get the following corollary in the spirit of the Zimmer’s problem

The proof of Theorem 2.5 relies on close, case by case examination of Thurston’s $3$-dimensional geometries, their finite volume quotients and their corresponding isometry groups. The computations of such groups can be summarized in the following

Main Theorem 2.5 is proved in Section 3 using the classification of isometry groups of orbifolds given by Theorem 2.8, together with rigidity results of semisimple Lie groups. To prove Theorem 2.8, one first need to understand finite volume quotients of Thurston’s three-dimensional geometries. Among such geometries, the most homogeneous ones are $S^{3}$, ${\mathbb{H}}^{3}$ and ${\mathbb{R}}^{3}$; and the remaining five present a more flexible description as fiber bundles

where $B$ is a two dimensional homogeneous geometry for $X$ either ${\mathbb{H}}^{2}\times{\mathbb{R}}$, $\widetilde{SL}_{2}({\mathbb{R}})$ or Nil and $B={\mathbb{R}}$ for $X$ either $Sol$ or $S^{2}\times{\mathbb{R}}$. In this context, a discrete group acting on the homogeneous space $X$, acts on the base space of the corresponding fiber bundle as well. The induced action on the base space of the fiber bundle presents a dual behaviour given by the following Theorem, whose proof can be derived from the discussions on [Thu97].

For the sake of completeness we present here a proof of Theorem 2.9. The proofs of Theorem 2.8 and Theorem 2.9 are carried out in a case by case setting on each Thurston geometry.

The structure of the paper is as follows: In Section 3 we present background material on three dimensional Alexandrov spaces, the Hilbert Smith conjecture and semisimple Lie groups and their lattices. Sections 5 through 13 cover the proof of Theorem 2.9 and Theorem 2.8 on each individual three dimensional geometry.

Theorem 2.9 can be used to obtain explicit characterizations of discrete groups acting on the corresponding three-dimensional geometry (in particular within the $Nil$ and $Sol$ cases), which lead to the proof of Theorem 2.8 in each case.

In this work we proposed a metric variation on the Zimmer program. The variation consisted in

• •

  Strengthening the category of automorphisms of the action from $C^{0}$ to isometries.

• •

  Relaxing the topological type of the spaces considered from smooth manifolds to Alexandrov spaces, which include three dimensional geometric orbifolds.

While Alexandrov spaces have an open dense subset which is a topological manifold, results related to Zimmer’s conjeture do not apply directly because the topological manifold is open, and the only rigidity results for actions on open manifolds in the spirit of the Zimmer program which are known to the authors are restricted to the one dimensional case [DH20]. In another instance of a complication, the manifold is not Riemannian, as Otsu-Shioya’s example shows [OS94].

There exist several instances of families of homeomorphisms of metric geodesic spaces for which rigidity results of actions of discrete groups can be proved. Among them we can consider, for a strengthening with respect to homeomorphisms and a weakening with respect to isometries:

• •

  Quasiconformal homeomorphisms, as in the alternative proof of Mostow Rigidity Theorem by [Bou09].

• •

  Bilipschitz homeomorphisms, as in the proof of the Hilbert-Smith Conjecture mentioned before [RS97].

• •

  Quasimöbius homeomorphisms as in [BK02], where rigidity results for them have as a consequence the rigidity of actions of quasi convex cocompact actions on ${\rm CAT(-1)}$-spaces.

Moreover, the specific analytic and geometric characteristics of the class of homeomorphisms are exploited in the process of proving an action rigidity result in an analogous way to how we used the geometric structure of Alexandrov three spaces in this work, inspired by the proofs of Zimmer program results in the diffeomorphism case.

Let us introduce the notation

for homeomorphisms of a metric space with a metric property $\alpha$ and let us refer to a metric condition $\alpha$ as a decoration in analogy with surgery theory, having at least the following examples in mind:

1. (i)

   The smooth case $\alpha={\rm Diff}$, for diffeomorphisms of the smooth structure associated to a Riemannian metric on a smooth manifold with fixed metric.

2. (ii)

   The topological case $\alpha={\rm Top}$, refering to homeomorphisms of a topological manifold.

3. (iii)

   The isometric case $\rm Iso$, meaning isometries of the geodesic metric space associated to a geometric three manifold, orbifold or Alexandrov space.

4. (iv)

   The quasiconformal case $\alpha=QC$, the referring to quasiconformal homemomorphisms, $\alpha=QM$, associated to quasimöbius homeomorphisms and $\alpha=BI$ for bilipschitz homemorphisms of a geodesic length metric space as discussed in the paragraph above.

We can consider the problem of describing the behaviour of a group homomorphism

to a homeomorphism group with decorations as described. Notice that $\alpha={\rm Diff}$ is the Zimmer program as described for instance in [Fis11],[Fis20], [Zim84]. On the other hand $\alpha={\rm Top}$ is the $C^{0}$-Zimmer program as described in [Wei11], [Ye20], [Ye19]. Finally $\alpha={\rm Iso}$ was the point of view adopted in this note, and $\alpha=QC,QM,BI$ are as before.

We would like to finish the present note with the following two questions:

• •

  To what extent a condition of prescribed curvature in metric spaces, such as the Alexandrov condition, or a choice of $QC$, $QM$ or $BI$-structures can be seen as a rigid structure, in the sense of Gromov [Fis11]?

• •

  For which decorations of homeomorphisms $\alpha$ is it possible to prove that homeomorphisms of a higher rank lattice $\Lambda$ with respect to the dimension of an Alexandrov space $X$

  $$\Lambda\longrightarrow{\rm AUT}^{\alpha}(X)$$

  or in general a metric measure space $X$ with finite Hausdorff dimension less than the rank of $\Lambda$ either factorize trough a finite quotient of $\Lambda$ or produce a quotient of infinite volume?

Recall that a closed three dimensional manifold $M$ is prime if it cannot be written as a connected sum with summands not homeomorphic to the three dimensional sphere.

By the prime decomposition Theorem, 3.15 page 31 in [Hem04], any closed three dimensional manifold can be written as a connected sum of prime factors.

Recall that a model geometry is a simply connected smooth three manifold $X$ together with a transitive action of a Lie group $G$ on $X$ with compact stabilizers such that $G$ is maximal among groups acting smoothly and transitively on $X$ with compact stabilizers.

A geometric structure on a three dimensional manifold $M$ is a diffeomorphism from $M$ to $X\diagup\Gamma$ for some model geometry $X$, where $\Gamma$ is a discrete subgroup of $G$ acting freely on X ;

It is a consequence of Thurston geometrization theorem due to Perelman, that any prime closed three dimensional manifold can be cut along three dimensional tori such that the interiors of the resulting manifold carries a geometric structure of finite volume.

The eight three dimensional geometries which admit at least one compact three manifold are :

1. (i)

   Spherical, where the three dimensional sphere is a representative.

2. (ii)

   Euclidean, where the flat three dimensional torus is an example,

3. (iii)

   Hyperbolic, where the three dimensional hyperbolic space is an example.

4. (iv)

   $\widetilde{SL}_{2}(\mathbb{R})$, where an example is given by the unit tangent bundle in a Riemannian metric of the tangent bundle over a genus two surface.

5. (v)

   $Nil$, where an example is given by the mapping torus of a Dehn twist on the two dimensional torus.

6. (vi)

   $Sol$, where an example is given by a manifold which fibers over the line with fiber the plane.

7. (vii)

   $S^{2}\times\mathbb{R}$.

8. (viii)

   $S^{1}\times\mathbb{H}^{2}$.

A closed 3-manifold has a geometric structure of at most one of the 8 types above.

We will need some preliminaries on three dimensional Alexandrov spaces. For a general reference see [BBI01].

For the purposes of this work, Alexandrov spaces are well behaved metric spaces which have three main properties that we want to highlight here:

1. (i)

   They have an open dense subset which is topological manifold.

2. (ii)

   Their isometry groups are Lie groups.

3. (iii)

   The class of Alexandrov spaces is stable under Gromov-Hausdorff convergence.

4. (iv)

   They include orbifolds over Riemannian manifolds.

In slightly more detail, Alexandrov spaces are a synthetic generalization of complete Riemannian manifolds with a lower bound on sectional curvature. The generalization uses comparison triangles with respect to the model spaces $S_{k}^{2}$, which are simply connected, two dimensional complete Riemannian manifolds of constant curvature $k$. More precisely, for $k>0$, $S_{k}^{2}$ is the sphere of radius $\frac{1}{\sqrt{k}}$, for $k<0$, $S_{k}^{2}$ is the hyperbolic plane $\mathbb{H}^{2}(\frac{1}{\sqrt{-k}})$ of constant curvature $k$, and for $k=0$, $S_{k}^{2}$ is the euclidean space $\mathbb{R}^{2}$.

Given a geodesic triangle in a geodesic length space $(X,d)$, with vertices $p,q,r\in X$, a comparison triangle in $S_{k}^{2}$ is a geodesic triangle $\bar{p}\bar{q}\bar{r}$ having the same side lengths. The geodesic length space $(X,d)$ is said to satisfy the Topogonov property for $k\in\mathbb{R}$, if for each triple $p,q,r\in X$ of vertices of a geodesic triangle, and each point $s$ on the geodesic from $q$ to $r$, the inequality $d(p,s)\geq d(\bar{p},\bar{s})$ holds, where $\bar{s}$ is the point on the geodesic side $\bar{q}\bar{r}$ of the comparison triangle with $d(\bar{p}\bar{s})=d(p,s)$.

Topogonov’s globalization theorem tells us that the local and global Toponogov property are equivalent in $k$-Alexandrov spaces. By Gromov’s precompactness theorem, Alexandrov $n$-dimensional spaces arise as Gromov-Hausdorf limits of compact riemannnian manifolds of dimension $n$ for which the sectional curvature is bounded below by $k$, and the diameter is bounded above by some fixed positive number $D$.

The class of $k$-Alexandrov spaces includes riemannian manifolds of sectional curvature bounded below by $k$, and several constructions including more general geodesic length spaces such as euclidean cones, suspensions, joins, quotients by isometric actions of compact Lie groups, and glueings along a submetry, see [GG16] section 2.2. From now on, we will omit the $k$ from the notation.

There exists a notion of angle between geodesics of an Alexandrov space, and a space of tangent directions at a given point $p$, denoted by $\Sigma_{p}$, can be defined as the completion of the metric space of equivalence classes of geodesics making a zero angle.

The space of tangent directions at a point $p$ in an Alexandrov space $X$, denoted by $\Sigma_{p}$, has the structure of a $1$-Alexandrov space of Hausdorf dimension ${\rm dim}(X)-1$. There is a set $R_{X}\subset X$, called the set of metrically regular points, where a point $p$ belongs to $R_{X}$ if its direction space $\Sigma_{p}$ is isometric to the radius one sphere. The complement is called the set of metrically singular points and denoted by $S_{X}=X\setminus R_{X}$. There are examples of Alexandrov spaces whose space of metrically singular points is dense, as seen in an example constructed in [OS94] as a limit of Alexandrov spaces, using baricentric subdivisions of a tetrahedron. However, for every Alexandrov space $X$, there is a dense subset of topologically regular points, whose space of directions are homeomorphic to a sphere (the set of topologically singular points is the complement of the set of topologically regular points). By Perelman’s conical neighborhood theorem, every point $p$ in an Alexandrov space has a neighborhood pointed homeomorphic to the euclidean cone over $\Sigma_{p}$, so that a locally compact, finite dimensional Alexandrov space has a dense subset which is a topological manifold.

In the specific case of dimension three, there are only two possibilities for the homeomorphic type of the space of directions, which is the two sphere $S^{2}$, for the topologically regular points and the real projective space $\mathbb{RP}^{2}$ for the topologically singular points. Let us summarize the basic structure of three dimensional Alexandrov spaces due to Galaz-García and Guijarro, compare Theorem 1.1 in page 5561 of [GGG15], and Theorem 3.1 and 3.2 in page 1196 of [GGG13]. See also [HS17].

A closed Alexandrov space is geometric if it can be written as a quotient of one of the eight geometries of Thurston under a cocompact lattice. The following theorem was proved as Theorem 1.6 in [GGG15] in page 5563. See also [HS17]

We now direct our attention to the isometry group of three dimensional Alexandrov spaces.

The following conjecture was formulated as an extension of Hilbert’s 5th Problem:

See [Tao14] for a modern account.

As a consequence of structural theorems of locally compact groups, such as the Gleason-Yamabe theorem and its predecessor by Von Neumann [vN33]. a counter-example to the Hilbert-Smith conjecture must contain a copy of a p-adic group $\widehat{\mathbb{Z}}_{p}$, for some $p$, see [Lee97], thus giving the equivalent conjecture

Conjecture 3.8 has been proven in different contexts. For example, if there is a notion of dimension which must be preserved, such as bi-Lipschitz actions of $\widehat{\mathbb{Z}}_{p}$ on Riemannian manifolds, where three notions of dimension coincide: Hausdorff dimension, cohomological dimension with integer coefficients and topological dimension. In such setting, the bi-Lipschitz condition tells us that the Hausdorff dimension on the quotient cannot decrease, but on the other hand a theorem by Yang [Yan60], tells us that the cohomological dimension of the quotient increases by two, leading to the following result:

The stronger setting of topological actions is much harder and has been proven only for small dimensions

Hence, the weaker version of the $p$-adic Hilbert-Smith conjecture for Alexandrov spaces holds, and we can consider the following conjecture:

A consequence of Theorem 3.10, gives us a three dimensional case of this result 14.4

A real Lie group is a Hausdorff topological group which is a $C^{\infty}$ smooth manifold for which the multiplication and inversion are smooth maps.

Recall that by Haar’s Theorem there exists up to a positive multiplicative constant, a unique countably additive, nontrivial measure $\mu$ on the Borel subsets of $G$ which is right translation invariant, has finite values on compact subsets, and is inner and outer regular.

The right translate $\mu(g^{-1}\quad)$by an element in the group $g^{-1}$ of a right invariant Haar measure $\mu$ is a right invariant Haar measure, and hence there exists a real function $\Delta$ satisfying

In the subsequent sections of this article, we will examine the isometry groups of three manifolds and their lattices, as well as the isometry groups of orbifolds or Alexandrov spaces.

If $\widetilde{X}$ is a complete, simply connected, Riemannian manifold and $\Gamma\subset Iso(\widetilde{X})$ a discrete subgroup of isometries, then $X/\Gamma$ has the structure of a complete Riemannian orbifold. The covering map $\rho:\widetilde{X}\rightarrow X$ satisfies the property that $\rho(x)=\rho(y)$ if and only if $\gamma x=y$ for some $\gamma\in\Gamma$. An isometry $\phi:X\rightarrow X$ lifts to $\widetilde{\phi}:\widetilde{X}\rightarrow\widetilde{X}$ such that $\rho\circ\widetilde{\phi}=\phi\circ\rho$ and for every $\gamma\in\Gamma$ and $x\in\widetilde{X}$ we have

thus there exist $\gamma^{\prime}\in\Gamma$ such that

that is $\widetilde{\phi}\circ\gamma\circ\widetilde{\phi}^{-1}=\gamma^{\prime}$ and $\widetilde{\phi}\in N=N_{Iso(\widetilde{X})}(\Gamma)$. This tells us that we have the isomorphism

Recall that an algebraic $\mathbb{R}$-group is a subgroup $\mathbb{G}_{\mathbb{C}}\subset GL_{m}(\mathbb{C})$ obtained as solutions of polynomial equations with coefficients over $\mathbb{R}$ and $\mathbb{G}_{\mathbb{R}}=\mathbb{G}_{\mathbb{C}}\cap GL_{m}(\mathbb{R})$ is a real Lie group. In this context we say that $\mathbb{G}_{\mathbb{R}}$ is a real form of $\mathbb{G}_{\mathbb{C}}$ or that $\mathbb{G}_{\mathbb{C}}$ is a complexification of $\mathbb{G}_{\mathbb{R}}$. The local structure of a Lie group is captured by its Lie algebra, so that two groups are locally isomorphic if and only if they have isomorphic Lie algebras, and thus, they can be obtained one from the other by taking connected components and topological covers.

The class of semisimple Lie groups can be defined as the class of Lie groups which are constructed up to covers and connected components from algebraic $\mathbb{R}$-groups which split as products $G_{1}\times\cdots\times G_{k}$, where each factor $G_{j}$ is simple. This definition is equivalent to other definitions of semisimple Lie groups aviailable in the literature, see [Zim84].

In the context of algebraic groups defined over a field $k$, the concept of $k$-rank is the maximal abelian subgroup which can be diagonalized over $k$. Thus, for a complex algebraic group, the $\mathbb{C}$-rank is the dimension of a maximal subgroup isomorphic to a complex torus $(\mathbb{C}^{*})^{l}$ and we are particularly interested in the real rank of a real form. We can observe that the real rank of a product $G_{1}\times\cdots\times G_{k}$ is the sum of the real rank of its factors $G_{j}$ and we can give some explicit examples.

Isotypic Lie groups are important because we can construct irreducible lattices in them, which don’t split as a product of lattices in the simple factors.

We notice that the classification of isometry groups presented here has as consequence classification results for finite group actions on three manifolds. There are three general kinds of behaviour:

• •

  Within the geometries whose isometry groups are extensions involving a discrete and finite volume subgroup of $SL_{2}(\mathbb{R})$, any finite group can act by isometries. This concerns the geometries $\mathbb{H}^{3}$, $\widetilde{SL_{2}}$ and $\mathbb{H}^{2}\times\mathbb{R}$.

• •

  For the case of spherical factors and the euclidean case, the classical results by Tollefson, Seifert-Threlfall and Bieberbach theorems exhaust the class of finite groups acting by isometries. This is recorded in [McC02], [Tol74], [Now34]. There exists a classification of free (topological) finite group actions in [LSY93], [HJKL02].

• •

  For Sol, any cyclic group can be realized as a consequence of the discussion of example 9.8. For the case Nil, 6.9 gives a classification depending on the results developed in this note.

  Notice that there exists a classification of topological free actions of finite groups on Nil and Sol manifolds based on the $p$- rank and P.A. Smith theory [Lee97], [LSY93].

In this short subsection we summarize the detailed chain of implications leading to the proof of Theorem 2.8.

Euclidean Geometry

According to the Bieberbach Theorems, there exists a discrete free abelian subgroup of translations $T$, which has rank less or equal to three. The assertion of theorem 2.8 for the isometry group of $R^{3}/\Gamma$ will be verified by examining the rank of the translation subgroup $T$, and discarding rank two and one by producing a quotient of infinite volume.

Nil Geometry.

For the Nil geometry, Theorem 2.8 is a consequence of 6.10, characterizing lattices of infinite volume, and Proposition 6.9 giving an exact sequence between isometry groups.

Spherical Geometry.

The theorem is a direct consequence of the classification of finite group actions on three -manifolds, as well as the determination of the components of the isometry groups 7.3. Notice that there are neither non-discrete subgroups of isometries nor groups whose quotient shows infinite volume within the spherical geometry.

$S^{2}\times\mathbb{R}$ Geometry.

The theorem is consequence of the splitting of the isometry groups, as well as the characterization of discrete subgroups of isometries in 8.3, finally concluding in 8.4.

Sol Geometry.

The theorem is stated as Corollary 9.7, which depends on the determination of centralizers in 9.6, and the determination of finite volume in 9.4.

Hyperbolic Geometry.

The theorem is direct consequence of Lemma 10.1, which is in turn consequence of Borel Density, or the preceding argumentation there. See also [Boc46].

$\mathbb{H}^{2}\times\mathbb{R}$ Geometry. The theorem is stated in 13.6, and it is consequence of Theorem12.1, stating the splitting of isometry groups of the factors.

$\widetilde{SL_{2}(\mathbb{R})}$ Geometry.

The Theorem is also stated in 13.6, and it is consequence of Lemma 10.1, and Proposition 13.5.

The classification of (topological) finite group actions on the torus by isometries has been concluded by work of Lee, Shin and Yokura [LSY93] and Ha, Jo, Kim and Lee [HJKL02].

It follows from the Bieberbach theorems that any topological action on the three torus is topologically conjugated to an isometry; moreover, by the fact that the three dimensional torus is sufficiently large in the sense of Heil and Waldhausen, [Wal68], any homotopy equivalence is homotopic to a homeomorphism, and any two homotopic homeomorphisms are isotopic.

Connected components

The isometry groups of co-compact euclidean orbifolds have been determined by Ratcliffe and Tschantz [RT15], in Theorem 1 and Corollaries 1 and 2 in pages 46 and 47, which we state now for later reference.

To understand why a compact quotient $\mathbb{R}^{3}/\Gamma$ could have as isometry group a torus of smaller dimension than $3$, we can take a look at two examples in dimension two:

If $\mathbb{F}$ is a commutative ring, denote by $H_{\mathbb{F}}$ the group of $3\times 3$ upper triangular matrices over $\mathbb{F}$ with $1$ in the diagonal, that is

The group $H_{\mathbb{R}}$ is a Lie group called the three dimensional Heisenberg group that fits into the exact sequence

where $\mathbb{R}\subset H_{\mathbb{R}}$ is its center. The three matrices

determine a canonical basis of the tangent space at the identity $T_{I}(H_{\mathbb{R}})$, so that its translations by left-multiplications gives us a basis of left invariant vector fields denoted by $X_{j}$ with $X_{j}(I)=e_{j}$. For a fixed element

the vector fields at $T_{g}(H_{\mathbb{R}})$ have expresions

If we consider the global coordinates

then a vector $v\in T_{g}(H_{\mathbb{R}})$ decomposes as

so that the left-invariant metric in $H_{\mathbb{R}}$ having $X_{j}(g)$ as an orthonormal basis is given in this coordinates as $ds^{2}=dx^{2}+dy^{2}+(dz-xdy)^{2}$. Being left-invariant, this metric has $H_{\mathbb{R}}$ as a subgroup of isometries given by left multiplication

for every $g\in H_{\mathbb{R}}$. Notice that there are other isometries that don’t come from left multiplication of $H_{\mathbb{R}}$. Such isometries form a group isometric to the orthogonal group $O(2)$ generated by the reflection $R(x,y,z)=(x,-y,-z)$ and the twisted rotations

where $\rho_{\theta}$ is a rotation in the $(x,y)$-plane with angle $\theta$, $\eta_{\theta}$ is a polynomial function in $x$ and $y$ and trigonometric in $\theta$.

The full isometry group $Iso(H_{\mathbb{R}})$ can be described as a semi-direct product $H_{\mathbb{R}}\rtimes O(2)$, because

meaning that the exact exact sequence

induced by the action on the quotient by the center $H_{\mathbb{R}}/\mathbb{R}\cong\mathbb{R}^{2}$ splits off, see [Sco83] for more details.

In this section we describe a series of ilustrative examples that capture the behaviour of every discrete subgroup of $Iso(H_{\mathbb{R}})$.