Distortion elements for surface homeomorphisms

TL;DR

This paper establishes a criterion involving the growth rate of diameters of iterates of a homeomorphism's lift to determine when such a homeomorphism is a distortion element in the group of surface homeomorphisms isotopic to the identity.

Contribution

It provides a new geometric condition based on diameter growth to identify distortion elements in the homeomorphism group of surfaces.

Findings

If the diameter growth condition is satisfied, then the homeomorphism is a distortion element.

The result links geometric growth rates to algebraic properties of the homeomorphism group.

The criterion applies to homeomorphisms isotopic to the identity on compact orientable surfaces.

Abstract

Let S be a compact orientable surface and f be an element of the group Homeo_{0}(S) of homeomorphisms of S isotopic to the identity. Denote by F a lift of f to the universal cover of S. In this article, the following result is proved: if there exists a fundamental domain D of the universal cover of S such that the sequence (d_{n}log(d_{n})/n) converges to 0 where d_{n} is the diameter of F^{n}(D), then the homeomorphism f is a distortion element of the group Homeo_{0}(S).