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Groups acting on products of locally finite trees

J. O. Button

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 $SL_2(\F_p(x,y))$ for any prime $p$.

Groups acting on products of locally finite 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 for any prime .
Paper Structure (8 sections, 13 theorems, 11 equations)

This paper contains 8 sections, 13 theorems, 11 equations.

Table of Contents

  1. Introduction
  2. Actions on locally finite trees
  3. Actions on a single tree
  4. Proper actions on products of trees
  5. Groups possessing proper actions
  6. Obstructions to acting properly on a product of trees
  7. Obstructions for hyperbolic groups
  8. Fields of positive characteristic and the Bruhat - Tits tree

Key Result

Proposition 2.5

The properties $(FA_{lf})$ and $(FA_{bv})$ are equivalent over the class of all finitely generated groups.

Theorems & Definitions (33)

  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • Example 2.4
  • Proposition 2.5
  • proof
  • Definition 2.6
  • Lemma 2.7
  • proof
  • Example 2.8
  • ...and 23 more