Characters of surface groups

TL;DR

This paper explores the space of characters of surface groups, showing it is a Poulsen simplex and that tracial representations can be approximated by factorial ones with spectral gap.

Contribution

It establishes that the space of traces of a surface group is the Poulsen simplex and introduces approximation results for tracial representations.

Findings

The space of traces of a surface group is the Poulsen simplex.

Any tracial representation can be approximated by factorial tracial representations with spectral gap.

Resolved a question posed by Orovitz, Slutsky, and the third author.

Abstract

We initiate the study of characters of surface groups and their corresponding tracial representations. We show that any tracial representation can be approximated arbitrarily well in the Wasserstein topology by factorial tracial representations with spectral gap. In particular, we deduce that the space of traces of a surface group is the Poulsen simplex, thereby resolving positively a question posed by Orovitz, Slutsky, and the third author.