Approximating stable translation lengths on fine curve graphs

TL;DR

This paper investigates the stable translation lengths of surface homeomorphisms on fine curve graphs, establishing their relation to finite approximations and revealing rationality properties of these lengths.

Contribution

It introduces a method to compare stable translation lengths of homeomorphisms with those of their finite approximations on curve graphs, and proves rationality of these lengths.

Findings

Stable translation length equals the supremum of finite approximations for homeomorphisms with dense periodic points.

Stable translation length is preserved under cell-like extensions.

Stable translation length of a mapping class on the curve graph is always rational.

Abstract

We study the stable translation length of homeomorphisms of a surface acting on the fine nonseparating curve graph and compare it to the stable translation lengths of its finite approximations - mapping classes relative to a finite invariant set - acting on the nonseparating curve graph. We prove that the stable translation length of a homeomorphism with a dense set of periodic points is the supremum of the stable translation lengths of its approximations, and that the stable translation length is preserved under cell-like extensions. We deduce that homotopically triv ial homeomorphisms of the torus have stable translation length which is the supremum of the stable translation lengths of their finite approximations. We show that the supremum is not always a maximum, by proving that the stable translation length of a mapping class acting on the nonseparating curve graph is rational.