Large-scale geometry of homeomorphism groups

Abstract

Let M be a compact manifold. We show the identity component 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.

Graphs of curves on infinite-type surfaces with mapping class group actions

“…This seems discouraging from the viewpoint of geometric group theory; however, there has been recent progress (see [36]) in studying the coarse geometry of topological groups that are neither finitely generated nor locally compact. A particularly beautiful application of this theory in [27] is the study of the large-scale geometry of homeomorphism groups of compact manifolds and its relation to the topology of and dynamics on the manifold.…”

Maximal pseudometrics and distortion of circle diffeomorphisms

“…In [9], Mann and Rosendal characterized the maximal metrics on groups of the form , which denotes the connected component of identity in the group of all homeomorphisms of a compact connected manifold — the maximal metrics are quasi‐isometric to the fragmentation metrics given by finite open coverings of . Earlier in [10], Militon studied distortion elements of , where is a compact surface, in terms of fragmentation length.…”

“…Let and let be a maximal pseudometric on . Adopting the definition proposed by Mann and Rosendal in [9], we will say that is distorted if the inclusion map is not a quasi‐isometric embedding, with respect to any right‐invariant Cayley metric on and with respect to on ; otherwise is undistorted . This definition does not depend on the choice of maximal pseudometric .…”

“…Earlier in [10], Militon studied distortion elements of , where is a compact surface, in terms of fragmentation length. By the results of [9], this approach is equivalent to studying distortion in the sense of maximal pseudometrics.…”

Strong distortion in transformation groups

“…This is particularly surprising since , as well as the groups for compact , are never strongly distorted, nor even strongly bounded, whenever . This is also true of provided that has infinite fundamental group — this follows from [, Example 6.8], or more explicitly from [, Proposition 20] which implies that all maximal metrics on are unbounded length functions. In particular, for these examples, there is no hope to improve Theorem to a proof of strong boundedness or distortion.…”