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Incommensurable lattices in Baumslag-Solitar complexes

Max Forester

Abstract

This paper concerns locally finite 2-complexes $X_{m,n}$ which are combinatorial models for the Baumslag-Solitar groups $BS(m,n)$. We show that, in many cases, the locally compact group Aut($X_{m,n}$) contains incommensurable uniform lattices. The lattices we construct also admit isomorphic Cayley graphs and are finitely presented, torsion-free, and coherent.

Incommensurable lattices in Baumslag-Solitar complexes

Abstract

This paper concerns locally finite 2-complexes which are combinatorial models for the Baumslag-Solitar groups . We show that, in many cases, the locally compact group Aut() contains incommensurable uniform lattices. The lattices we construct also admit isomorphic Cayley graphs and are finitely presented, torsion-free, and coherent.
Paper Structure (15 sections, 6 theorems, 11 equations)

This paper contains 15 sections, 6 theorems, 11 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Graphs
  4. $G$--trees
  5. Automorphisms of CW complexes
  6. Locally finite complexes
  7. Lattices
  8. Generalized Baumslag--Solitar groups
  9. Definitions
  10. Fibered $2$--complexes
  11. Non-elementary GBS groups
  12. Segments and index
  13. The modular homomorphism
  14. The orientation character
  15. Labeled graphs and deformations

Key Result

Theorem 1.1

If $\gcd(k,n) \not= 1$ then $\mathop{\mathrm{Aut}}\nolimits(X_{k,kn})$ contains uniform lattices $G_1, G_2$ that are not abstractly commensurable.

Theorems & Definitions (13)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1
  • Example 3.2
  • Example 3.3
  • Lemma 3.6
  • Lemma 3.8
  • proof
  • Remark 3.10
  • Lemma 3.11
  • ...and 3 more