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Tiling in some nonpositively curved groups

Joseph MacManus, Lawk Mineh

Abstract

We prove that acylindrically hyperbolic groups are monotileable. That is, every finite subset of the group is contained in a finite tile. This provides many new examples of monotileable groups, and progress on the question of whether every countable group is monotileable. In particular, one-relator groups and many Artin groups are monotileable.

Tiling in some nonpositively curved groups

Abstract

We prove that acylindrically hyperbolic groups are monotileable. That is, every finite subset of the group is contained in a finite tile. This provides many new examples of monotileable groups, and progress on the question of whether every countable group is monotileable. In particular, one-relator groups and many Artin groups are monotileable.
Paper Structure (9 sections, 26 theorems, 40 equations)

This paper contains 9 sections, 26 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Quasiisometries and quasigeodesics
  4. Hyperbolic spaces
  5. Acylindrical actions
  6. Fixed points at infinity
  7. Swingers and tiles
  8. Finding swingers in acylindrically hyperbolic groups
  9. Main results

Key Result

Theorem 1

Word-hyperbolic groups are monotileable.

Theorems & Definitions (53)

  • Theorem : Akhmedov
  • Theorem 1
  • Corollary 2
  • Corollary 3
  • Lemma 2.1
  • proof
  • Lemma 2.2: dructu2018geometric
  • Lemma 2.3: dructu2018geometric
  • Lemma 2.4
  • Theorem 2.5: Osin_Acyl_hyp
  • ...and 43 more