4 Universality phenomena
The term universality loosely describes the following situation:
There is a class
dvipng error! exitcode was 2 (signal 0), transscript follows: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
dvipng error! exitcode was 2 (signal 0), transscript follows: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
dvipng error! exitcode was 2 (signal 0), transscript follows:hyperbolic buildings in Example \ref{bldgs} are all homeomorphic (since they are all homeomorphic to the stable Menger space
dvipng error! exitcode was 2 (signal 0), transscript follows:).
3. Let
dvipng error! exitcode was 2 (signal 0), transscript follows:be a closed
dvipng error! exitcode was 2 (signal 0), transscript follows:-manifold,
dvipng error! exitcode was 2 (signal 0), transscript follows:be its triangulation. Then
dvipng error! exitcode was 2 (signal 0), transscript follows:determines a right-angled Coxeter graph
dvipng error! exitcode was 2 (signal 0), transscript follows:and
dvipng error! exitcode was 2 (signal 0), transscript follows:-dimensional David-Vinberg complex
dvipng error! exitcode was 2 (signal 0), transscript follows:. We assume, in addition, that
dvipng error! exitcode was 2 (signal 0), transscript follows:is a flag-complex, satisfying the no-square condition (which guarantees hyperbolicity of the resulting Coxeter group).
Suppose
dvipng error! exitcode was 2 (signal 0), transscript follows:are two such triangulations of
dvipng error! exitcode was 2 (signal 0), transscript follows:, which admit a common subdivision. Let
dvipng error! exitcode was 2 (signal 0), transscript follows:. Then (H.~Fischer MR1941443):
dvipng error! exitcode was 2 (signal 0), transscript follows:
Note that
dvipng error! exitcode was 2 (signal 0), transscript follows:
In particular, the boundaries which appear in case
dvipng error! exitcode was 2 (signal 0), transscript follows:are of three types:
dvipng error! exitcode was 2 (signal 0), transscript follows:, 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
dvipng error! exitcode was 2 (signal 0), transscript follows:is a topological invariant of
dvipng error! exitcode was 2 (signal 0), transscript follows:?
Problem 4.3 (Comments) [Misha Kapovich, Tadeusz Januszkiewicz] Suppose that
dvipng error! exitcode was 2 (signal 0), transscript follows:and
dvipng error! exitcode was 2 (signal 0), transscript follows:are closed
dvipng error! exitcode was 2 (signal 0), transscript follows:-manifolds equipped with flag-triangulations, so that
dvipng error! exitcode was 2 (signal 0), transscript follows:
Does it follow that every prime connected sum summand of
dvipng error! exitcode was 2 (signal 0), transscript follows:appears as a connected sum summand of
dvipng error! exitcode was 2 (signal 0), transscript follows:,
dvipng error! exitcode was 2 (signal 0), transscript follows:? What can be said in higher dimensions?