Boundaries/Section3 - Problems in Geometric Group Theory

3 Boundaries of Coxeter groups

Let $G$ be a finitely-generated Coxeter group with Coxeter presentation $\<S|R\>$ . This presentation determines a Davis-Vinberg complex $X$ (see MR2360474), whose dimension equals rank of the maximal finite special subgroup of $G$ with respect to the above presentation. The complex $X$ admits a natural piecewise-Euclidean $CAT(0)$ metric. The group $G$ acts on $X$ properly discontinuously and cocompactly. Hence, $X$ has visual boundary $\D X$ , which we can regard as a boundary of $G$ . Topology of $\D X$ was studied in MR1684267, MR1804695. For instance, MR1684267 constructs examples of hyperbolic Coxeter groups whose boundaries are both orientable and non-orientable Pontryagin surfaces and 2-dimensional Menger compacta. Recall that a Pontryagin surface is obtained as follows. Let $K$ be a connected, compact (without boundary) triangulated surface. Define $P(K)$ by replacing each closed 2-simplex $\si$ in $K$ with a copy $K_\si$ of the closure of $K\setminus \si$ . We get the map

$P(K)\to K$

by sending each $K_\si$ to $\si$ . Set $P_n:= P(P_{n-1})$ . Then the corresponding Pontryagin surface $P_{\infty}$ based on $P_0$ is inverse limit of the sequence

$.... P_n \to P_{n-1} \to ... \to P_1\to P_0.$

It turns out that $P_{\infty}$ can have only three distinct topological types:

If $P_0\cong S^2$ , then $P_{\infty}\cong S^2$ .
If is oriented but has genus $\ge 1$ , then $P_\infty$ is oriented (i.e. $H^2(P_\infty, \Z)\cong \Z$ ) but not homeomorphic to .
If is not oriented then $P_\infty$ is unoriented. In this case, the rational homological dimension of $P_\infty$ equals 1.

Problem 3.1 (Comments) [Alexander Dranishnikov] Is it true that isomorphic Coxeter groups have homeomorphic boundaries?

Remark It appears that the answer is positive provided that all labels are powers of $2$ . REFERENCE?

Problem 3.2 (Comments) [Alexander Dranishnikov] Does there exist a Coxeter group $G_n$ with $n$ -dimensional boundary $\D G_n$ , so that the rational homological dimension of $\D G_n$ equals $1$ ?

Problem 3.3 (Comments) [Alexander Dranishnikov] Under which conditions on the Coxeter diagram of $G$ , the boundary of a Coxeter group is $n$ -connected and locally $n$ -connected?

Partial results in this direction are obtained in MR1422863. The main motivation for this problem comes from the problem of realizing Menger spaces as boundaries of Coxeter groups.

Problem 3.4 (Comments) [Misha Kapovich] Can exotic homology manifolds as in MR1394965 appear as ideal boundaries of Coxeter groups?

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