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Strong shortcuts, generating sets, and isometric circles in asymptotic cones

Nima Hoda, Timothy Riley

Abstract

We show that whether loops can be shortcut in a group's Cayley graph depends on the choice of finite generating set. Our example is the direct product of two rank-2 free groups and a consequence is that this group has asymptotic cones with isometrically embedded circles that are null-homotopic.

Strong shortcuts, generating sets, and isometric circles in asymptotic cones

Abstract

We show that whether loops can be shortcut in a group's Cayley graph depends on the choice of finite generating set. Our example is the direct product of two rank-2 free groups and a consequence is that this group has asymptotic cones with isometrically embedded circles that are null-homotopic.

Paper Structure

This paper contains 2 sections, 5 theorems, 6 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Proof

Key Result

Theorem 1.1

The Cayley graph $\textup{Cay}(G, A)$ of the product of two rank-2 free groups is strongly shortcut when $A = \{a,b,c,d\}$ but not when $A = \{a,b,c,db^{-1}\}$.

Figures (2)

  • Figure 1: Left: the four defining relators of the presentation \ref{['pres']} for $G$. Right: a "ziggurat" van Kampen diagram for the word $w_5 = t^5 c t^{-5}a t^5 c^{-1}t^{-5}a^{-1}$.
  • Figure 2: The van Kampen diagram $D$ for the word $u_kv^{-1}$.

Theorems & Definitions (12)

  • Theorem 1.1
  • Corollary 1.2
  • Lemma 2.1
  • proof
  • Claim 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 2 more