Surface groups, infinite generating sets, and stable commutator length

TL;DR

This paper provides a new proof that the Cayley graph of a surface group, with certain generating sets, has infinite diameter, highlighting the group's large-scale geometric complexity.

Contribution

It offers a novel proof of Calegari's theorem regarding the infinite diameter of Cayley graphs for surface groups with specific generating sets.

Findings

Cayley graph of surface groups has infinite diameter with certain generating sets

The result applies to generating sets including all simple closed curves

Provides a new proof of an existing theorem

Abstract

We give a new proof of a theorem of D. Calegari that says that the Cayley graph of a surface group with respect to any generating set lying in finitely many mapping class group orbits has infinite diameter. This applies, for instance, to the generating set consisting of all simple closed curves.