Conjugacy class of homeomorphisms and distortion elements in groups of homeomorphisms

TL;DR

This paper characterizes non-spreading homeomorphisms on surfaces with boundary, linking them to conjugacy close to identity or rotations, and relates non-spreading to distortion in the group of homeomorphisms.

Contribution

It establishes a precise equivalence between non-spreading and conjugacy close to identity or rotation, and characterizes distortion elements in groups of homeomorphisms of surfaces.

Findings

Non-spreading homeomorphisms are conjugate to elements close to identity or rotation.

On certain surfaces, non-spreading elements are exactly the distorted elements.

The results extend known theorems to a broader class of surfaces with boundary.

Abstract

Let S be a compact connected surface and let f be an element of the group Homeo\_0(S) of homeomorphisms of S isotopic to the identity. Denote by \tilde{f} a lift of f to the universal cover of S. Fix a fundamental domain D of this universal cover. The homeomorphism f is said to be non-spreading if the sequence (d\_{n}/n) converges to 0, where d\_{n} is the diameter of \tilde{f}^{n}(D). Let us suppose now that the surface S is orientable with a nonempty boundary. We prove that, if S is different from the annulus and from the disc, a homeomorphism is non-spreading if and only if it has conjugates in Homeo\_{0}(S) arbitrarily close to the identity. In the case where the surface S is the annulus, we prove that a homeomorphism is non-spreading if and only if it has conjugates in Homeo\_{0}(S) arbitrarily close to a rotation (this was already known in most cases by a theorem by B{\'e}guin, Crovisier, Le Roux and Patou). We deduce that, for such surfaces S, an element of Homeo\_{0}(S) is distorted if and only if it is non-spreading.