Examples of Homologically Wide Actions

Abstract

An action of a finite group G on a manifold M is homologically wide if the first homology of the manifold contains the regular representation of the group. In this chapter, we study this notion independently of the rest of this monograph. We first study the case where M is a surface and G acts freely, using the Lefschetz fixed point formula. We then recall a result of Broughton on the homology representation that allows us to deal with the case of a general action on a Riemann surface. After this, we switch to higher dimensional manifolds. We first recall Curtis’s theory of the virtual Lefschetz characters, and use results of Cooper and Long to construct examples and counterexamples to homological wideness in all dimensions ≥ 3. Finally, we study locally symmetric spaces. Property (T) implies that only the case of rank 1 is interesting, where we make some remarks on the relation to automorphic representations and Mostow rigidity. Finally, we study an example of using torsion homology: Mednykh’s explicit computation of the homology representation of the Seifert-Weber dodecahedral space, which we identify with a known modular representation through Brauer characters.

An action of a finite group G on a manifold M is homologically wide if the first homology of the manifold contains the regular representation of the group. In this chapter, we study this notion independently of the rest of this monograph. We first study the case where M is a surface and G acts freely, using the Lefschetz fixed point formula. We then recall a result of Broughton on the homology representation that allows us to deal with the case of a general action on a Riemann surface. After this, we switch to higher dimensional manifolds. We first recall Curtis’s theory of the virtual Lefschetz characters, and use results of Cooper and Long to construct examples and counterexamples to homological wideness in all dimensions ≥ 3. Finally, we study locally symmetric spaces. Property (T) implies that only the case of rank 1 is interesting, where we make some remarks on the relation to automorphic representations and Mostow rigidity. Finally, we study an example of using torsion homology: Mednykh’s explicit computation of the homology representation of the Seifert-Weber dodecahedral space, which we identify with a known modular representation through Brauer characters.

1 Surfaces

In dimension 2, the situation for a fixed-point free action is clear because the homology representation can be computed using the Lefschetz fixed point theorem [48, Theorem 2C.3].

Proposition 9.1.1

A fixed-point free action of a non-trivial finite group G on a closed orientable surface M is Q -homologically wide if and only if M is hyperbolic (i.e., has negative Euler characteristic). In particular, the property is independent of G.

Proof

We will compute the character χ _h of the rational homology representation

$$\displaystyle \begin{aligned} h=h_{\mathbf{Q}} \colon G \rightarrow \operatorname{H}_1(M,\mathbf{Q}). \end{aligned}$$

First of all, since we assume any g ≠ e has no fixed points, the map g: M → M has Lefschetz number 0, i.e.,

$$\displaystyle \begin{aligned} \operatorname{tr}(g_* | \operatorname{H}_0(M,\mathbf{Q})) - \operatorname{tr}(g_* | \operatorname{H}_1(M,\mathbf{Q})) + \operatorname{tr}(g_* | \operatorname{H}_2(M,\mathbf{Q})) = 0. \end{aligned}$$

Since the action of g _∗ on $\operatorname {H}_0(M,\mathbf {Q}) \cong \mathbf {Q}$ and $\operatorname {H}_2(M,\mathbf {Q}) \cong \mathbf {Q}$ is trivial (induced by the action of g on the space of connected components of M, respectively the 2-cells, i.e., the one-element sets), both outer terms in this expression are 1, and the middle term is χ _h(g) by definition, so we find that χ _h(g) = 2 for g ≠ e. For g = e, on the other hand, we get directly from the definition of the character that $\chi _h(e) = \operatorname {tr}(e_* | \operatorname {H}_1(M,\mathbf {Q})) = \dim \operatorname {H}_1(M,\mathbf {Q})=b_1(M) = 2-\chi _M.$ We conclude that

$$\displaystyle \begin{aligned} \chi_h(g) = \left\{ \begin{array}{ll} 2 \$ \mbox{if } g \neq e, \\ 2 - \chi_M \$ \mbox{if } g=e. \end{array} \right. \end{aligned}$$

On the other hand, the character of the regular representation is

$$\displaystyle \begin{aligned} \chi_{{G,\mathrm{reg}}} = \left\{ \begin{array}{ll} 0 \$ \mbox{if } g \neq e, \\ |G| \$ \mbox{if } g=e. \end{array} \right. \end{aligned}$$

We can match these expressions, and since representations are isomorphic if and only if their characters are equal, we find

$$\displaystyle \begin{aligned} h = 2 \cdot {\mathbf{1}}_G - \frac{\chi_M}{|G|} \cdot \rho_{G,\mathrm{reg}}, \end{aligned}$$

and hence the regular representation of a non-trivial group G occurs inside h if and only if χ _M < 0. □

The above proposition has no (constant) curvature assumption. In constant curvature but with more general actions, we have the following.

Proposition 9.1.2

Any action of a finite group G by (not necessarily fixed-point free) conformal automorphisms on a closed Riemann surface M is Q -homologically wide if χ _G∖M < 0.

Proof

We rely on the computation of the character of h in this branched setting by Broughton in [20, Prop. 2(iii)], using in addition (taking into account the holomorphic structure) the Eichler trace formula. Suppose that the G-cover is branched above t points. For each branch point, choose a lift to the cover and let C _i ≤ G denote the (cyclic) stabiliser of that lift (the stabilisers of any lift of a given point are conjugate in G). Then

$$\displaystyle \begin{aligned} h \$ = 2 \cdot \mathbf{1} - \chi_{G \backslash M} \cdot \rho_{G,\mathrm{reg}} + \sum_{i=1}^t (\rho_{G,\mathrm{reg}}- \operatorname{Ind}_{C_i}^G \mathbf{1} ) \\ \$ = 2 \cdot \mathbf{1} - \chi_{G \backslash M} \cdot \rho_{G,\mathrm{reg}} + \sum_{i=1}^t \operatorname{Ind}_{C_i}^G (\rho_{C_i,\mathrm{reg}}-\mathbf{1} ), \end{aligned} $$

where we have used that the induced representation of the regular representation of C _i to G is the regular representation of G. Since 1 occurs in $\rho _{C_i,\mathrm {reg}}$, the representations occurring in the sum are not virtual (i.e., every irreducible representation of G occurs in it with non-negative multiplicity) and therefore h contains ρ _G,reg as soon as χ _G∖M < 0. □

Remark 9.1.3

In the “non-orbifold quotient” setting of Proposition 9.1.1, by the Riemann-Hurwitz formula, χ _M < 0 if and only if χ _G∖M < 0. This is no longer true in the setting of Proposition 9.1.2, when we only have χ _G∖M < 0 ⇒ χ _M < 0 but not the other way around.

The following is a detailed version of Corollary 1.2.3.

Corollary 9.1.4

Let M ₁, M ₂ be two commensurable non-arithmetic closed Riemann surfaces. Then they admit a diagram ( 1.1 ) and, assuming the corresponding orbifold M ₀ satisfies $\chi _{M_0} < 0$ , isometry of M ₁ and M ₂ can be checked by computing the multiplicity of zero in at most

$$\displaystyle \begin{aligned} 4 ((\chi_{M_1}\chi_{M_2}/(\chi^{\mathrm{orb}}_{M_0})^2)!)^2 \end{aligned}$$

twisted Laplace spectra, where $\chi ^{\mathrm {orb}}_{M_0}$ is the orbifold Euler characteristic given by

$$\displaystyle \begin{aligned} \chi^{\mathrm{orb}}_{M_0}:=\chi_{M_0} - \sum (1-1/n_i), \end{aligned} $$

(9.1)

with n _i the order of the stabiliser group at the orbifold points.

Proof

By Proposition 2.5.2, hyperbolic non-arithmetic commensurable closed Riemann surfaces automatically admit a diagram of the form (1.1), and by Proposition 9.1.2, every group action is Q-homologically wide since we assume $\chi _{M_0}<0$. Therefore, Theorem 1.2.1 applies. To find a prime number ℓ coprime to |G|, we can always choose ℓ > |G|, and by Bertrand’s postulate [47, Thm. 418], we can find such a prime ℓ ≤ 2|G|. Hence we can make the bound in Theorem 1.2.1 weaker by $2 \ell |H_2^{\mathrm {ab}}| \leq 4 |G|{ }^2$. To express this entirely in terms of the original diagram, we set d _i to be the degree of M _i → M ₀. Notice that the compositum $M_1 \bullet _{M_0} M_2$ is of degree at most d ₁ d ₂ over M ₀, and the degree of the normal closure of the compositum is of degree at most (d ₁ d ₂)! over M ₀ (see (2.3)). Hence |G|≤ (d ₁ d ₂)!. Now $d_i = \chi _{M_i}/\chi ^{\mathrm {orb}}_{M_0}$ where $\chi ^{\mathrm {orb}}_{M_0}$ is the orbifold Euler characteristic given by $\chi _{M_0} - \sum (1-1/n_i)$ for n _i the order of the stabiliser group at the orbifold points [25, 5.1.3]. We find an upper bound of at most

$$\displaystyle \begin{aligned} 4 ((d_1 d_2)!)^2 \leq 4 ((\chi_{M_1}\chi_{M_2}/(\chi^{\mathrm{orb}}_{M_0})^2)!)^2 \end{aligned}$$

for the number of equalities of multiplicities that needs to be checked. □

Remark 9.1.5

If M ₁ and M ₂ as in Corollary 9.1.4 are isospectral, by Weyl’s law, they have the same volume. Since d _i = vol(M _i)∕vol(M ₀), we can then assume that d ₁ = d ₂ and $\chi _{M_1} = \chi _{M_2}$.

In Chap. 11, one finds some detailed examples of surfaces with less crude bounds on the required number of equalities.

2 Using the Virtual Lefschetz Character

The above arguments in dimension 2 are based on very precise information given by fixed point formulæ. These admit a generalisation to a setup as in diagram (1.2) with M of arbitrary dimension, where they can sometimes be used to deduce some information about condition (∗∗) from Sect. 7.1; more specifically, whether $\operatorname {Ind}_H^G \mathbf {1}$ is a subrepresentation of the homology representation (over Q). For this, we use that our manifolds are closed, and thus admit a regular triangulation, which allows us to apply the work of Curtis [32]. Consider the virtual Lefschetz character of G given as

$$\displaystyle \begin{aligned} \Lambda(g)= -h_{\mathbf{Q}}(g) + \sum_{i \neq 1} (-1)^i \operatorname{tr}(g_* | \operatorname{H}_i(M,\mathbf{Q})). \end{aligned}$$

Then by [32, Prop. 1.6], we have

$$\displaystyle \begin{aligned} \langle \operatorname{Ind}_H^G \mathbf{1}, \Lambda \rangle = \chi(M_1), \end{aligned} $$

(9.2)

the Euler characteristic of M ₁. The formula provides no information for 3-dimensional manifolds, since then χ(M ₁) = 0 and Λ = 0. The next proposition provides an example of a result that can be deduced from such methods.

Proposition 9.2.1

if M is of dimension 4 and χ(M ₁) ≤ 0, then $\operatorname {Ind}_H^G \mathbf {1}$ and h _Q have at least one irreducible representation in common.

Proof

By Poincaré duality, $\operatorname {H}_3(M,\mathbf {Q}) = \operatorname {Hom}(\operatorname {H}_1(M,\mathbf {Q}),\mathbf {Q}) \cong \operatorname {H}_1(M,\mathbf {Q})$, hence

$$\displaystyle \begin{aligned} \Lambda = 2 \cdot \mathbf{1} + \operatorname{H}_2(M,\mathbf{Q}) - 2 h_{\mathbf{Q}}, \end{aligned}$$

and we conclude from (9.2) that

$$\displaystyle \begin{aligned} 2 \langle \operatorname{Ind}_H^G \mathbf{1}, h_{\mathbf{Q}} \rangle = 2 + \langle \operatorname{Ind}_H^G \mathbf{1}, \operatorname{H}_2(M,\mathbf{Q}) \rangle - \chi(M_1) \geq 2 - \chi(M_1) > 0. \end{aligned}$$

Such results do not suffice to completely verify whether condition (∗∗) holds under general topological conditions in higher dimension (i.e., only referring to the vector space structure of $\operatorname {H}_1(M,\mathbf {Q})$, and not to its Q[G]-module structure), and indeed, in the next few sections we will see examples showing that this is not possible.

3 Manifolds of Dimension ≥ 3

In case of 3-manifolds, the picture can vary widely: it is possible to construct a class of closed 3-manifolds with Q-homologically wide group actions, but also hyperbolic 3-manifolds with large isometry group for which only the trivial group action is Q-homologically wide. This upgrades to similar results in higher dimensions. The results are direct consequences of the work of Cooper and Long [27] for topological manifolds.

Proposition 9.3.1

Suppose N is a smooth compact connected 3-manifold with a free smooth action by a finite group G, and let γ denote a smooth simple closed curve in N such that the orbit G ⋅ γ consists of |G| disjoint smooth simple closed curves. Let

$$\displaystyle \begin{aligned} X=N-\mathscr{N}(G \cdot \gamma) \end{aligned}$$

denote the open manifold N with an open regular neighbourhood $\mathscr {N}(G \cdot \gamma )$ of the G-orbit of γ removed, and let M denote the double of the manifold X (i.e., two copies of X glued together along their boundaries). Assume that the embedding ι: ∂X → X of the boundary induces a surjective map on first homology groups $\iota _* \colon \operatorname {H}_1(\partial X,\mathbf {Q}) \rightarrow \operatorname {H}_1(X,\mathbf {Q}).$ Define a Riemannian metric on M as the pullback of any Riemannian metric on the quotient manifold G∖M; then if M′ is any (n − 3)-dimensional closed smooth connected Riemannian manifold with trivial G-action, M × M′ is an n-dimensional closed smooth connected Riemannian manifold with a free isometric G-action that is Q -homologically wide.

Proof

Since [27] concerns topological manifolds, we start by observing that the double of a smooth manifold has a smooth structure compatible with the embedding of the original manifold, but composed with a diffeomorphism on one of the copies, see, e.g., [58, VI.5]. Hence M is a smooth compact connected manifold on which the finite group G acts smoothly (and properly) without fixed points. Therefore, the quotient G∖M is a smooth compact connected manifold, too, and the quotient map M → G∖M is a smooth covering map. Choose a Riemannian structure on the quotient G∖M such that it becomes a closed Riemannian manifold, and make M into a closed Riemannian manifold by giving it the pullback Riemannian structure. Now G acts on M by fixed-point free Riemannian isometries.

Cooper and Long [27, Lemma 2.3] have proven that, by the assumption that ι _∗ is surjective,

$$\displaystyle \begin{aligned} \operatorname{H}_1(M, \mathbf{Q}) = \rho_{G,\mathrm{reg}} \oplus (\rho_{G,\mathrm{reg}}-\mathbf{1}) \end{aligned}$$

as G-modules. Hence, in particular, $\operatorname {H}_1(M, \mathbf {Q})$ contains the regular representation.

Now G acts trivially on M′, so G acts by isometries on the cartesian product M × M′. By the Künneth formula, M × M′ has first homology group

$$\displaystyle \begin{aligned} \operatorname{H}_1(M \times M', \mathbf{Q}) = \operatorname{H}_1(M,\mathbf{Q}) \oplus \operatorname{H}_1(M',\mathbf{Q}), \end{aligned}$$

so ρ _G,reg is also a subrepresentation of $\operatorname {H}_1(M \times M', \mathbf {Q})$, and the action of G on M × M′ is Q-homologically wide. □

In the other direction, Cooper and Long have also shown that through Dehn surgeries, it is possible to “remove” the canonical Q[G]-modules ρ _G,reg and also ρ _G,reg−1 from the homology representation to arrive at a rational homology 3-sphere with a G-action. Further surgery along an embedded hyperbolic knot allows one to construct such a hyperbolic (i.e., constant − 1 curvature) manifold [27, Theorem 2.6].

Proposition 9.3.2

For any finite non-trivial group G, there exists a hyperbolic rational homology 3-sphere M with a free action of G by isometries on M; in particular, the action of G on M is not Q -homologically wide.

We conclude that in dimension 3 homological wideness is unrelated to hyperbolicity (in marked contrast to the case of dimension 2).

Corollary 9.3.3

For any finite non-trivial group G, and any dimension n ≥ 3, there exists an n-dimensional closed connected Riemannian manifold M′ with a free action of G by isometries on M′ for which the action of G on M′ is not Q -homologically wide.

Proof

Let M be as in Proposition 9.3.2, and let G act trivially on the (n − 3)-dimensional sphere S ⁿ⁻³. Setting M′ = M × S ⁿ⁻³, by the Künneth formula, we have $\operatorname {H}_1(M',\mathbf {Q}) = 0$ for n ≠ 4 and $\operatorname {H}_1(M',\mathbf {Q}) = \mathbf {Q}$ for n = 4, so it is impossible for non-trivial G to act homologically wide on M′. □

Remark 9.3.4

Bartel and Page have shown that there exists a closed hyperbolic 3-manifold M with a free action of any given finite group G by isometries on M such that additionally, $\operatorname {H}_1(M,\mathbf {Q})$ is any given Q[G]-module [7].

4 Locally Symmetric Spaces of Rank ≥ 2

Let G denote a connected semisimple Lie group with trivial center, K a maximal compact subgroup of G, and Γ a discrete subgroup of G such that Γ∖G is compact. Consider the locally symmetric Riemannian manifold M :=  Γ∖G∕K. If all factors of G have real rank ≥ 2, then Γ has Kazhdan’s property (T), and hence $\operatorname {H}_1(M,\mathbf {Q})= \Gamma ^{\mathrm {ab}} \otimes _{\mathbf {Z}} \mathbf {Q} = \{0\}$ (see, e.g., [8, Cor. 1.3.6]). This shows the following.

Proposition 9.4.1

Only the trivial group can have a Q -homologically wide action on a locally symmetric space of rank ≥ 2.

5 Locally Symmetric Spaces of Rank 1

On the other hand (keeping the notations of Sect. 9.4), if G has rank 1, the first Betti number of M can be expressed in terms of representation theory via a formula of Matsushima’s [67]; more precisely, a sum of multiplicities of specific representation occurring in the representation R _Γ of G by right multiplication on L ²( Γ∖G). If G = SO(n, 1) for n ≥ 3, there is a unique representation J ₁ in that sum and b ₁(M) equals the multiplicity of the representation J ₁ in R _Γ. Here, J ₁ is the unique unitary irreducible representation with non-zero Lie algebra cohomology. Except for n = 3, J ₁ is not in the discrete or principal series ([34, Thm. V.5; Rem. V.8; Prop. V.6]; [50, Lemma 4.4] or [14, VII.4.9]). For n = 3, J ₁ is the principal series representation of PSL(2, C) on L ²(C) with Gelfand-Graev-Vilenkin parameters (2, 0), given explicitly as

$$\displaystyle \begin{aligned} J_1\left(\left(\begin{smallmatrix}a \$ b \\ c \$ d \end{smallmatrix} \right)\right) (f)(z):= (cz+d)^2 f\left(\frac{az+b}{cz+d}\right). \end{aligned}$$

Proposition 9.5.1

Let $M=\Gamma \backslash \mathbb {H}^n$ denote a closed hyperbolic n-manifold (n ≥ 3), corresponding to a cocompact discrete subgroup Γ in SO(n, 1). If G is a finite group acting Q -homologically widely on M, then

$$\displaystyle \begin{aligned} |G| \leq \langle J_1, R_\Gamma\rangle, \end{aligned}$$

the multiplicity of the representation J ₁ described above in the SO(n, 1)-representation R _Γ given by right multiplication on L ²( Γ∖SO(n, 1)).

6 Hyperbolic Manifolds; Formulation in Terms of Uniform Lattices

If $M=\Gamma \backslash \mathbb {H}^n$ is a compact connected hyperbolic manifold of dimension n ≥ 3 with finite full isometry group Isom(M), Mostow rigidity implies that Isom(M)≅Out( Γ), the outer automorphism group of Γ; indeed, M is an Eilenberg-MacLane K( Γ, 1), and hence Out( Γ) is isomorphic to the group of homotopy self-equivalences up to free homotopy; but by Mostow rigidity, every homotopy equivalence is homotopic to an isometry [72, Thm. 24.1’]. Hence homological wideness of the action of a subgroup G↪Isom(M) on M can be formulated in purely group theoretical terms.

Proposition 9.6.1

The action of a finite group G of isometries on a compact connected hyperbolic manifold $M=\Gamma \backslash \mathbb {H}^n$ of dimension n ≥ 3 is K-homologically wide if and only if the representation

$$\displaystyle \begin{aligned} G \hookrightarrow \mathrm{Out}(\Gamma) \rightarrow \mathrm{Aut}(\Gamma^{\mathrm{ab}} \otimes_{\mathbf{Z}} K) \cong \mathrm{GL}(b_1(\Gamma),K) \end{aligned}$$

contains the regular representation.

Recall again that Proposition 9.3.2 gives an example where this representation is trivial for K = Q. Belolipetsky and Lubotzky [9] have shown that, given any finite group G, there exist infinitely many compact connected hyperbolic manifolds with G as isometry group.

Remark 9.6.2

Besson et al. [12, Théorème 9.1] have proven that homeomorphic oriented hyperbolic manifolds of the same dimension n ≥ 3 with the same volume are isometric. In our situation, we start from a “correspondence” as in diagram (1.1) and a homeomorphism is not given.

7 Using Torsion Homology for Homological Wideness

Instead of using Q-homological wideness, one may try to find suitable ℓ for which the action of G on $\operatorname {H}_1(M)$ is F _ℓ-homologically wide for specific ℓ. For example, $\operatorname {H}_1(M)$ might be torsion, so that no non-trivial group acts Q-homologically wide, but nevertheless, $\operatorname {H}_1(M,{\mathbf {F}}_\ell )$ can contain F _ℓ[G]. We content ourselves with commenting on one example.

Example 9.7.1

Let M denote the Seifert–Weber dodecahedral space, a hyperbolic 3-manifold with first Betti number zero; cf. [98]. By Mostow rigidity, M is uniquely described by its fundamental group

$$\displaystyle \begin{aligned} \langle a_1,\dots,a_6 | a_3^{-1} a_6 a_4^{-1} a_5 a_2, \$a_2^{-1} a_6 a_3^{-1} a_4 a_1, a_6 a_2^{-1} a_3 a_5 a_1^{-1},\\ \$a_2a_4 a_5^{-1}a_6 a_1^{-1}, a_3 a_4^{-1} a_6 a_5^{-1} a_1, a_4 a_2 a_5 a_3 a_1 \rangle. \end{aligned} $$

For the following facts, especially the computation of the homology representation, we refer to Mednykh [68]:

• $\operatorname {H}_1(M,\mathbf {Z}) = {\mathbf {F}}_5^3$, admitting non-trivial maps to a cyclic group Z∕ℓ Z for ℓ = 5.

• The full isometry group of M is isomorphic to S ₅. If we write generators as r = (12) and c = (12345), there is a faithful action on $\operatorname {H}_1(M,{\mathbf {F}}_\ell )$ through matrices in GL(3, F ₅) given as

  $$\displaystyle \begin{aligned} r = \left( \begin{matrix} 4 \$ 2 \$ 4 \\ 0 \$ 0 \$ 2 \\ 0 \$ 3 \$ 0 \end{matrix}\right), \ c = \left( \begin{matrix} 0 \$ 1 \$ 0 \\ 0 \$ 0 \$ 1 \\ 1 \$ 2 \$ 3 \end{matrix}\right). \end{aligned}$$

The isometry group G = 〈r〉≅Z∕2Z of M has a cyclic vector (1, 1, 0) in $\operatorname {H}_1(M,{\mathbf {F}}_\ell )$, so the action of G on M is F ₅-homologically wide (but not Q-homologically wide). Similarly, the isometry group G = 〈crc ⁻¹ r〉≅Z∕3Z of M has cyclic vector (1, 0, 0). $\lozenge$

We can identify this homology representation on the nose, as follows. Let

$$\displaystyle \begin{aligned} \mathrm{sgn} \colon S_5 \rightarrow {\mathbf{F}}_5^* \end{aligned}$$

denote the linear character given by the sign of a permutation (modulo 5), and let

$$\displaystyle \begin{aligned} \psi \colon S_5 \rightarrow \operatorname{GL}(3,{\mathbf{F}}_5) \end{aligned}$$

denote the 3-dimensional irreducible representation constructed as follows as composition factor of the standard permutation representation of S ₅. The group S ₅ acts on $V:={\mathbf {F}}_5^5$ by permuting the standard basis vectors; consider the quotient W = V∕L by the S ₅-invariant line L := F ₅ ⋅ (1, 1, 1, 1, 1) spanned by the all-one vector, and consider the S ₅-invariant hyperplane

$$\displaystyle \begin{aligned} U:=\{ [w=(w_1,\dots,w_5)] \in W \colon \sum w_i = 0 \} \end{aligned}$$

in W (this makes sense since we work modulo 5). Then the natural induced action of S ₅ on $U \cong {\mathbf {F}}_5^3$ is the 3-dimensional faithful mod-5 representation ψ, that turns out to be irreducible.

Proposition 9.7.2

The mod-5 homology representation of the isometry group S ₅ of the Seifert–Weber dodecahedral space is irreducible, and can be identified with

$$\displaystyle \begin{aligned} \rho \cong \mathrm{sgn} \otimes \psi. \end{aligned} $$

(9.3)

Proof

Let ρ: S ₅ →GL(3, F ₅) denote the explicit realisation of the representation, as in Example 9.7.1. To see the isomorphism in (9.3), we compute that the values of the Brauer character of ρ on the conjugacy classes of elements of order coprime to 5 (given by cycle type) are as in Table 9.1.

Table 9.1 Brauer character of the mod-5 homology representation for the Seifert–Weber dodecahedral space

This equals the Brauer character of sgn ⊗ ψ, so ρ has the same semi-simplification, but since sgn ⊗ ψ is irreducible, it is actually isomorphic to ρ, that is then also automatically irreducible. □

Remark 9.7.3

If M is a closed hyperbolic 3-manifold with large $\dim _{{\mathbf {F}}_\ell } \operatorname {H}_1(M,{\mathbf {F}}_\ell )$, there are sometimes explicit lower bounds on the volume of M, expounded in works of Culler and Shalen. For example, if for some prime ℓ, $\dim _{{\mathbf {F}}_\ell } \operatorname {H}_1(M,{\mathbf {F}}_\ell ) \geq 5$ then vol(M) > 0.35 [31].

Remark 9.7.4

The scope of the method of using ℓ-torsion in $\operatorname {H}_1(M)$ in establishing homological wideness (or the weaker conditions we outlined) is unclear, although some heuristics can be set up by considering the size of the (ℓ-)torsion subgroup. Bader et al. [5, Theorem 1.8] have constructed, for any α ≥ 0, sequences {M _m} of (non-arithmetic) closed hyperbolic rational homology 3-spheres, converging in Benjamini–Schramm topology, for which

$$\displaystyle \begin{aligned} |\operatorname{H}_1(M_m)_{\mathrm{tors}}| \sim e^{\alpha\, \mathrm{vol}(M_m)}. \end{aligned} $$

(9.4)

A conjecture of Bergeron and Venkatesh [10, Conjecture 1.3 for SO(3, 1), and bottom of page 122] states that if $M=\Gamma \backslash \mathbb {H}^3$ is closed hyperbolic arithmetic manifold and $M_n:=\Gamma _m \backslash \mathbb {H}^3$ for a chain of congruence subgroups of Γ_m of Γ with trivial intersection, then the growth in (9.4) holds with α = 1∕(6π).

Project

Explicitly determine the isometry group of rational homology 3-spheres and its representation on the torsion in the first homology. Which (modular) representations can occur? (cf. Remark 9.3.4 for the free part.)

Open Problem

In the situation of Remark 9.7.4, understand not just the size, but also (some of) the decomposition of $\operatorname {H}_1(M_m,{\mathbf {F}}_\ell )$ as a Z[Out( Γ_m)]-module. A subproblem is, given a hyperbolic manifold $\Gamma \backslash \mathbb {H}^n$ corresponding to a cocompact discrete group Γ of isometries of hyperbolic n-space $\mathbb {H}^n$ (n ≥ 3), to determine the structure of its first homology Γ^ab as a Z[Out( Γ)]-module (compare Sect. 9.6).

Project

Develop other topological criteria on manifolds that imply homological wideness for large classes of group actions.