Long Colimits of Topological Groups III: Homeomorphisms of Products and Coproducts

The group of compactly supported homeomorphisms on a Tychonoff space can be topologized in a number of ways, including as a colimit of homeomorphism groups with a given compact support or as a subgroup of the homeomorphism group of its Stone-Čech compactification. A space is said to have the Compactly Supported Homeomorphism Property (CSHP) if these two topologies coincide. The authors provide necessary and sufficient conditions for finite products of ordinals equipped with the order topology to have CSHP. In addition, necessary conditions are presented for finite products and coproducts of spaces to have CSHP.

UNIVERSAL MINIMAL FLOWS OF GENERALIZED WAŻEWSKI DENDRITES

We study universal minimal flows of the homeomorphism groups of generalized Ważewski dendrites WP, $P \subseteq \left\{ {3,4, \ldots ,\omega } \right\}$. If P is finite, we prove that the universal minimal flow of the homeomorphism group H (WP) is metrizable and we compute it explicitly. This answers a question of Duchesne. If P is infinite, we show that the universal minimal flow of H (WP) is not metrizable. This provides examples of topological groups which are Roelcke precompact and have a nonmetrizable universal minimal flow with a comeager orbit.

Large-scale geometry of homeomorphism groups

Let $M$ be a compact manifold. We show that the identity component $\operatorname{Homeo}_{0}(M)$ of the group of self-homeomorphisms of $M$ has a well-defined quasi-isometry type, and study its large-scale geometry. Through examples, we relate this large-scale geometry to both the topology of $M$ and the dynamics of group actions on $M$. This gives a rich family of examples of non-locally compact groups to which one can apply the large-scale methods developed in previous work of the second author.