Groups acting on products of locally finite trees

TL;DR

This paper investigates group actions on products of trees, explores hyperbolic surface groups' actions, and explicitly embeds a genus 2 surface group into a matrix group over a function field.

Contribution

It provides new insights into which groups act on products of trees and offers an explicit embedding of a hyperbolic surface group into a matrix group.

Findings

Hyperbolic surface groups likely act properly on finite products of trees.

Explicit embedding of genus 2 surface group into SL_2 over a function field.

Evidence supporting hyperbolic surface groups' actions on products of trees.

Abstract

We examine the question of which finitely generated groups act properly on a finite product of locally finite simplicial trees and present evidence in favour of hyperbolic surface groups having such an action. We also give a completely explicit embedding of the genus 2 closed hyperbolic surface group in $SL_2(\mathbb{F}_p(x,y))$ for any prime $p$.