More hyperbolic spaces with isomorphic boundary C*-algebras

In this blog post, we are going to show that neither the fundamental group, nor the corresponding action on the geodesic boundary on a hyperbolic manifold, are uniquely determined by the boundary C*-algebra. We will finish the post by an outlook of possible paths for the non-commutative version of the Mostow rigidity theorem discussed in a previous post.

This is the second post, showing a negative answer to the question of whether the boundary C*-algebras ${C(\partial X)\rtimes_r \Gamma}$ for a (torsion-free) lattice $\Gamma$ determines the underlying locally symmetric space $\Gamma \backslash X$ up to isometry. Here we find lattices ${\Gamma, \Gamma' \subset \H_{\mathbb R}^3}$ for which

${C(\partial X)\rtimes_r \Gamma = C(\partial X)\rtimes_r \Gamma'} \quad \text{but} \quad {\Gamma \neq \Gamma'}$

as abstract groups.

This yeilds examples of topologically non-equivalent dynamical systems of amenable actions of lattices on the ${3}$ -sphere whose C*-algebra is isomorphic. It also diminishes the hope to find a non-commutative version of the classical Mostow rigidity theorem using the boundary algebras ${C(X)\rtimes_r \Gamma}$ as opposed to the boundary dynamical systems ${(\partial X, \Gamma)}$ to classify the closed locally symmetric spaces ${\Gamma \backslash \H_{\mathbb R}^n}$ for $n\geq 3$ . The previous blog post introduced the basic ideas and objects we will be working with here so I assume some familiarity with that post.

Let us start by stating the main theorem which this blog post will prove:

Proposition 1 – There are non-isomorphic lattices ${\Gamma, \Gamma'\subset SO^0(3,1)}$ which are torsion-free and cocompact, for which

$C(\partial \H_{\mathbb R}^3)\rtimes_r \Gamma = C(\partial \H_{\mathbb R}^3)\rtimes_r \Gamma'.$

Let us recall what the main ingredients of the above proposition are. The space ${\partial \H_{\mathbb R}^3}$ denotes the geodesic boundary of the real hyperbolic 3-space. This is homeomorphic to a 2-sphere. It is constructed by adding one point at infinity for every (unit speed) geodesic ray emanating from a (or any) fixed point ${x_0\in \H_{\mathbb R}^n}$ . The union ${\overline{\H}_{\mathbb R}^3 := \H_{\mathbb R}^n \sqcup \partial \H_{\mathbb R}^n }$ can be topologized in such a way that the action of ${Iso(\H_{\mathbb R}^3)}$ extends to the boundary ${\partial \H_{\mathbb R}^3}$ . This is called the geodesic compactification (or Gromov compactification, or visual compactification). All this is covered in great detail in the book of Lizhen Ji and Armand Borel “Compactifications of symmetric and locally symmetric spaces”.

We will call ${C(\partial \H^3_{\mathbb R})\rtimes_r \Gamma}$ the boundary C*-algebras. These were introduced in a previous blog post and are of course the C*-algebras associated with the transformation groupoid ${\partial \H_{\mathbb R}^3\rtimes_r \Gamma}$ which are shown in this article to be Kirchberg algebras in the UCT class. One of the best features of these algebras is that they are classified by their K-theory.

Note that all that has been said so far remains valid if we replace ${\H_{\mathbb R}^3}$ with any rank 1 symmetric space of non-compact type.

Notation: Let us denote by ${G}$ the group ${SO^0(3,1)}$ and ${K = SO(n) \subset G}$ the maximal compact subgroup. The standard realization of ${\H_{\mathbb R}^3 = G/K}$ as a symmetric space can be found in for instance this article. The groups ${\Gamma, \Gamma' \subset G}$ will always denote torsion-free cocompact lattices. We denote ${P\subset G}$ a fixed (minimal) parabolic subgroup of ${G}$ and ${P= MAN}$ its Langlands decomposition. Note that since ${P}$ is minimal, ${M = P\cap K}$ .

We will need to assume ${\Gamma \backslash G/K}$ admits a spin ${^\C}$ -structure. This is a rather mild condition that I will show is satisfied in our case, and holds in general whenever ${G}$ is simply connected (which it is not in our case though). Then we have the following –

Lemma 2 – Assume that ${\Gamma \backslash G / K}$ admits a spin ${^{\mathbb C}}$ -structure. Then there is a KK-equivalence after a shift of parity by ${1}$

$C(\partial \H_{\mathbb R}^3) \rtimes_r\Gamma \sim_{KK} C(\Gamma \backslash G /M).$

Proof

Since ${G}$ has rank 1 we have a ${G}$ -equivariant isomorphism ${ G/P= K/M = \partial \H^3_{\mathbb R}}$ . The first equality follows from the Iwasawa decomposition: ${G/P = KAN/MAN = K/M}$ . Now using the (strong) Morita equivalence ${C(G/P) \rtimes_r\Gamma \sim_M C_0(\Gamma \backslash G) \rtimes_r P}$ , we can exploit the the fact that ${P}$ is solvable (being a Borel subgroup), hence amenable. Then Corollary 2.7.5 of Blackadar’s “K-Theory for Operator Algebras”, gives us a KK-equivalence up to shift of parity by ${dim(G/P) ~ mod(2) = 1}$ to ${C(\Gamma\backslash G) \rtimes_r M}$ where ${M\subset P}$ is the maximal compact of ${P}$ (this is where we need a spin ${^{\mathbb C}}$ -structure).

Now since ${\Gamma}$ is torsion-free, we have that the action of ${M\subset K}$ is free, hence we get a Morita equivalence

$C(\Gamma \backslash G)\rtimes_r M = C(\Gamma \backslash G/ M)$

which concludes the proof.

Thus ${C(G/P)\rtimes_r \Gamma = C(G /P)\rtimes_r \Gamma'}$ for two torsion-free cocompact lattices (assuming a spin ${^{\mathbb C}}$ -structure) if and only if ${\Gamma \backslash G / K}$ and ${\Gamma'\backslash G / K}$ have the same ${K}$ -theory, or equivalently if they are KK-equivalent, being in the bootstrap class. This simplifies the work considerably as we may connect the isomorphism class of these algebras with the K-theory of the space ${\Gamma \backslash G / M}$ .

It can be shown that ${\Gamma \backslash G /M}$ is nothing but the unit tangent bundle of ${\Gamma \backslash G /K}$ , and thus of some interest in its own right, as it determines the geodesic flow. Let us tie it to the K-theory of the locally symmetric space ${\Gamma \backslash G /K }$ :

Lemma 3 – There is an isomorphism of ${{\mathbb Z}_2}$ -graded abelian groups

$K^*(\Gamma \backslash G / M) = K^*(\Gamma \backslash G / K)\otimes {\mathbb Z}^2$

Proof

The bundle

$G/P = K/M \rightarrow \Gamma \backslash G / M \rightarrow \Gamma \backslash G / K$

gives us an Atiyah-Hirzebruch spectral sequence

$H^p(B\Gamma, K^q(K/M)) \Rightarrow K^{p+q}(\Gamma \backslash G / M)$

Similarly, the K-theory of ${\Gamma \backslash G /K}$ is determined by the trivial bundle

$\{pt\}\rightarrow \Gamma \backslash G / K\rightarrow \Gamma \backslash G /K$

So when ${n = dim(G/K)}$ is odd or equivalently when $K/M$ is even, the two spectral sequences are related by a simple tensoring with ${{\mathbb Z}^2}$ .

Note that if ${G}$ was of rank ${n}$ then if ${P_0 = M_0A N_0}$ is a minimal parabolic subgroup of ${G}$ the above lemma would still hold true and give us an isomorphism

$K^\star(\Gamma \backslash G / M_0) = K^\star(\Gamma \backslash G / K) \otimes {\mathbb Z}^{|\Delta|}$

where ${|\Delta|}$ is the cardinality of the set of (restricted) roots of ${G}$ (see this article for the K-theory of $G/P_0$ )

It is known that there are more than one arithmetic hyperbolic homology ${3}$ -spheres. One being the Brieskorn $\Sigma(-2, 3,7)$ manifold, the other can be obtained by surgery on the three-twist knot (see this video lecture). Consequently, there exists two cocompact, arithmetic lattices ${\Gamma, \Gamma' \subset G}$ such that

$\Gamma \backslash G/K \qquad \text{ and } \qquad \Gamma' \backslash G / K$

are homology ${3}$ -spheres. We are going to assume only that the locally symmetric spaces are orientable as this can be done without loss of generality. To see this, assume they are not orientable, we can pass to a sublattice $\hat{\Gamma}\subset \Gamma$ of index two (hence automatically normal) corresponding to the oriented double cover. This is clearly also an arithmetic torsion-free cocompact lattice. Then using the identity (which follows from an application of Lyndon’s spectral sequence)

$H^i(\hat{\Gamma}, \Z) = H^i(\Gamma, Hom_\Gamma(\hat{\Gamma}, \Z))$

we can conclude that also the $\hat{\Gamma}$ is a homology 3-sphere, since the right hand side of the above group is torsion free, it admits only the trivial $\Z_2$ -action.

For 3-manifolds orientability implies (by Wu’s formula) that the second Stiefel-Whitney class vanishes. Hence we can assume without loss of generality that our manifolds are orientable. Well, strictly speaking, we need to know that there are at least three lattices which are homology spheres, so that we don’t pass from one to the other when passing to the oriented double cover.

Now since $H^\star(\Gamma \backslash G /K)$ and $H^\star(\Gamma' \backslash G /K)$ are torsion free, the Atiyah-Hirzebruch spectral sequence of these spaces collapses on the second page, and we get that

$K^i(\Gamma \backslash G / K) = K^i(\Gamma' \backslash G / K) = \Z \quad \text{for }i= 0, 1.$

We can thus use the above lemmas to conclude that

$C(\partial \H_{\mathbb R}^3)\rtimes_r \Gamma = C(\partial \H_{\mathbb R}^3)\rtimes_r \Gamma'$

while ${\Gamma \neq \Gamma'}$ . This concludes the proof of Proposition 1.

It follows that the boundary algebras cannot in these cases distinguish between the homotopy type of the associated locally symmetric space (even if we restrict to arithmetic cocompact lattices) or equivalently, by Mostow’s rigidity theorem, their isometry class.

By a result of Emery, there are no integral (or even rational) hyperbolic homology n-spheres for ${n \geq 5}$ (source) so there may still be hope to salvage some kind rigidity in higher dimensions. Curiously, there are no homology spheres that are of “simplest type” in the terminology of Vinberg and Shvartsman, which could be another class of lattices worth investigating further.

For now however the takeaway is that as the boundary algebras, being are essentially determined by the cohomology of the associated locally symmetric space, seem to be too coarse an invariant to distinguish the symmetric spaces, at least in the rank 1 case.

Leave a Reply Cancel reply