Bounding surface actions on hyperbolic spaces

TL;DR

This paper establishes a diameter bound for fundamental domains of surface groups acting isometrically on hyperbolic spaces, linking geometric properties like genus and injectivity radius to the action's domain size.

Contribution

It provides a new diameter bound for fundamental domains based on hyperbolicity, genus, and injectivity radius, advancing understanding of surface group actions on hyperbolic spaces.

Findings

Diameter bound depends on delta, genus, and injectivity radius

Bound applies to isometric actions of surface groups on hyperbolic spaces

Results connect geometric group theory with hyperbolic geometry

Abstract

We give a diameter bound for fundamental domains for isometric actions of the fundamental group of a closed hyperbolic surface on a delta-hyperbolic space, where the bound depends on the hyperbolicity constant delta, the genus of the surface, and the injectivity radius of the action, which we assume to be strictly positive.