Character varieties and actions on products of trees

TL;DR

This paper investigates conditions under which surface groups can act freely and properly on finite products of bounded valence trees, introducing arithmetic criteria and methods involving character varieties over positive characteristic fields.

Contribution

It provides new obstructions and an arithmetic criterion for free and proper actions of surface groups on products of bounded valence trees, linking geometric actions to algebraic character varieties.

Findings

Obstructions to actions on bounded valence trees identified

An arithmetic criterion for existence of such actions established

Methods for studying character varieties over positive characteristic fields developed

Abstract

It is well known that surface groups admit free and proper actions on finite products of infinite valence trees. In this note, we address the question of whether there can be a free and proper action on a finite product of bounded valence trees. We provide some obstructions and an arithmetic criterion for existence. The bulk of the paper is devoted to an approach to verifying the arithmetic criterion by studying the character variety of certain surface groups over fields of positive characteristic. The methods may be useful for attempting to determine when groups admit good linear representations in other contexts.