Hyperbolic surfaces with isomorphic boundary C*-algebras

Given a manifold $\overlien{X}$ with boundary $\partial X$ , and isometry $G$ , for each subgroup $\Gamm$ of $G$ one can form the crossed product C*-algebra $C(\partial X)\rtimes_r\Gamma$ . This post is a one in a series of posts that investigates the non-commutative analogues of the Mostow rigidity theorem, which shows that for certain locally symmetric spaces the fundamental group is a complete isomorphism (i.e. isometry) invariant for the locally symmetric spaces, setteling the Borel conjecture for such spaces.

In this post we look at the failure of the theorem in the special case of $SL_2(\R)$ .

In what follows, let $G = SL_2(\R)$ , $K = SO_2(\R)\subset G$ (a maximal compact subgroup), and $\Gamma\subset G$ be any torsion free cocompact lattice, meaning $G/\Gamma$ compact. We have already covered Mostow’s rigidity theorem in an earlier blog post. For our purposes we will write it as follows:

Suppose that $\Gamma\backslash X$ is irreducible and X is not equal to the hyperbolic plane $\H^2$ . If $\Gamma\backslash X$ and $\Gamma'\backslash X'$ are homotopic, then they are isometric when the metrics on $X$ and $X′$ are suitably scaled on their reducible factors.

For the rank 1 case (i.e. the case where a(ny) maximal flat submanifold of $X$ has dimension $1$ ), the proof relies on the construction of a $\Gamma$ -equivariant homeomorphism $\partial X \to \partial X'$ , where $\partial$ denotes the geodesic boundary, which then descends to an isometry (up to scaling) of the locally symmetric spaces $\Gamma\backslash X$ and $\Gamma'\backslash X'$ .

So the geometry of the locally symmetric space $\Gamma\backslash X$ is completely determined by the isomorphism class of the dynamical system $(\partial X, \Gamma)$ of the action of $\Gamma$ on the geodesic boundary of $X$ .

Constructing the crossed product C*-algebra $C(\partial X)\rtimes_r\Gamma$ , henceforth referred to as the boundary C*-algebra, usually results in loss of information, which means one cannot expect it to be a complete invariant of the isomorphism type of the underlying locally symmetric space.

Mostow’s rigidity theorem does not hold for $G = SL_2(\R)$ . The very modest goal of this post is proving the same also does not hold for the boundary C*-algebras, that is, there are isomorphic boundary C*-algebras associated to non-isometric locally symmetric spaces for $G$ . This is necessarily true as they constitute a strictly finer family of invariants then the boundary dynamical systems, but for the sake of the exposition let see why.

Proposition – There are lattices $\Gamma, \Gamma' \subset G$ for which $\Gamma\backslash G / K$ and $\Gamma'\backslahs G/ K$ are non-isometric, but for which $C(\partial (G/K)\rtimes_r \Gamma = C(\partial (G/K)\rtimes_r \Gamma')$

In fact there are uncountably many such lattices, as we will see shortly. Note that $\Gamma\backslash G / K$ and $\Gamma'\backslash G / K$ are isometric if and only if $\Gamma$ and $\Gamma '$ are conjugate. The above proposition will be an obvious consequence of the following results:

The isomorphism type of $C(\partial (G/K)\rtimes_r \Gamma$ is determined by its K-theory: This can be shown as follows. Since $G$ has rank 1, $G$ acts transitively on the boundary of $G/K$ an the stabilizer subgroup $P$ of any point on the boundary is a minimal parabolic subgroup. Hence $\partial G/K = G/P \simeq S^1$ . We use the strong Morita equivalence $C(G/P)\rtimes_r \Gamma \sim C(\Gamma \backslahs G)\rtimes_r P$ of Rieffel (see this article) together with the fact that since $P = NA\ltimes (M)$ is minimal it is amenable being a compact extension of a solvable Lie group $NA$ where $P = MAN$ is the Langlands decomposition of $P$ (see this book), hence the crossed product $C(\Gamma \backslash G)\rtimes_r P$ is a Kirchberg algebra (see Prop. 3.4 of this article). It follows that $C(G/P)\rtimes_r \Gamma$ is also a Kirchberg algebra. By classification theory, Kirchberg algebras are uniqeuly determined by their K-theory.

The K-theory of $C(\partial (G/K)\rtimes_r \Gamma$ is uniquely determined by the genus of $\Gamma \backslash G/K$ : Let $g$ be the genus, then for instance by this article the K-theory is determined (on p. 35) to be

$K_0(\partial G/K)\rtimes \Gamma) = \Z^{2g +1}\oplus \Z/\langle 2g - 2 \rangle \qquad K_1(\partial G/K)\rtimes \Gamma) = \Z^{2g +1}.$

For any genus $g\geq 2$ there are lattices $\Gamma, \Gamma' \subset G$ for which $\Gamma \backslash G/K$ and $\Gamma' \backslash G/K$ are closed non-isometric locally symmetric spaces of genus $g$ : This is a classical result: For $g\geq 2$ the space of conjugacy classes of torsion free cocompact lattices of $G$ has the structure of a manifold of dimension $(6g-6)$ (see Proposition 5 of this article with $n = 0$ ). As mentioned above the conjugacy classes of lattices correspond to isometry classes of the corresponding locally symmetric spaces.

This concludes the proof.

It may at this point be tempting to try to salvage the above rigidity theory by restricting our attention only to “arithmetic” cocompact lattices. Arithmetic lattices are sometimes defined to be any subgroup commensurable with $PSL_2(\Z)$ (the integer points of the algebraic group $PSL_2(\R)$ ). None of these lattices will be cocompact however, since $PSL_2(\Z)$ is not. Loosely speaking, using different representations of $PSL_2(\R)$ one can produce different “integer points” for $PSL_2(\R)$ , some of which will be cocompact. These will be our cocompact arithmetic lattices. For the precise definition see this wonderful book by Morris.

Given a cocompact torsion free arithmetic lattice $\Gamma \subset PSL_2(\R)$ the associated hyperbolic closed surface $\Gamma \backslash PSL_2(\R)/ PSO_2(\R)$ will be called an arithmetic surface. Using the formula of this article we have that as $g\to \infty$ , the number of isomorphism classes of arithmetic surfaces of genus $g$ (denoted here by $AL(g)$ ) satisfies the following limit behaviour

$\lim_{g\to \infty} \frac{log(AL(g))}{glog(g)} = 2$

as $glog(g) = log(g^g) \to \infty$ faster than $O(g)$ , we can deduce that $log(AL(g))$ must go to infinity faster than $O(g)$ too. Hence $AL(g) \to \infty$ at a rate faster than $e^g$ , hence (way) faster than $O(g)$ . It follows that for sufficiently large $g$ “most” genera will have an abundance of associated arithmetic cocompact lattices. Repeating the above argument, we have thus proved the following

Corollary – There are arithmetic cocompact lattices $\Gamma, \Gamma' \subset G$ for which $\Gamma\backslash G / K$ and $\Gamma'\backslahs G/ K$ are non-isometric, but for which $C(\partial (G/K)\rtimes_r \Gamma = C(\partial (G/K)\rtimes_r \Gamma')$ .

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