Boundaries/Section4 - Problems in Geometric Group Theory

4 Universality phenomena

The term universality loosely describes the following situation:

There is a class ${\mathcal C}$ of groups (spaces) of different nature, whose ideal boundaries are all homeomorphic.

Usually such results come from topological rigidity results for certain families of compacta.

Examples of universality phenomena.

1. Consider the class of all 2-dimensional hyperbolic groups which are 1-ended, do not split over virtually cyclic groups, are not commensurable to surface groups, are not relative $PD(3)$ groups. Then the ideal boundaries of all groups in this class are homeomorphic to the Menger curve. See MR1834498.

2. The boundaries of the right angled rank $n+1$ hyperbolic buildings in Example \ref{bldgs} are all homeomorphic (since they are all homeomorphic to the stable Menger space $M_{n,k}$ ).

3. Let $N$ be a closed $n$ -manifold, $\Del$ be its triangulation. Then $\Del$ determines a right-angled Coxeter graph $Cox(N,\Del)$ and $n+1$ -dimensional David-Vinberg complex $C(N, \Del)$ . We assume, in addition, that $\Del$ is a flag-complex, satisfying the no-square condition (which guarantees hyperbolicity of the resulting Coxeter group).

Suppose $\Del_1, \Del_2$ are two such triangulations of $N$ , which admit a common subdivision. Let $C_i:= C(N, \Del_i)$ . Then (H.~Fischer MR1941443):

$\geo C_1 = \geo C_2.$

Note that

$\geo C(N, \Del)= \geo C(N\# N, \Del\# \Del).$

In particular, the boundaries which appear in case $n=2$ are of three types: $S^2$ , oriented Pontryagin surface, non-orientable Pontryagin surface.

Problem 4.1 (Comments) [Tadeusz Januszkiewicz] Find more universality phenomena.

Problem 4.2 (Comments) [Misha Kapovich] Is it true that $\geo C(N, \Del)$ is a topological invariant of $N$ ?

Problem 4.3 (Comments) [Misha Kapovich, Tadeusz Januszkiewicz] Suppose that $(N_1, \Del_1)$ and $(N_2, \Del_2)$ are closed $3$ -manifolds equipped with flag-triangulations, so that

$\geo C(N_1, \Del_1)= \geo C(N_2, \Del_2).$

Does it follow that every prime connected sum summand of $N_i$ appears as a connected sum summand of $N_{i+1}$ , $i=1,2$ ? What can be said in higher dimensions?

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