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Dyer groups have the falsification by fellow-traveller property

Megan Howarth

Abstract

This paper is devoted to the study of the falsification by fellow-traveller property (FFTP) in Dyer groups. We exhibit a finite generating set for which the associated Cayley graph is a locally finite mediangle graph, and leverage its properties to prove that Dyer groups have the FFTP. It follows that Dyer groups have finitely many cone types, emphasising their role in providing a unified approach to Coxeter groups and graph products of cyclic groups.

Dyer groups have the falsification by fellow-traveller property

Abstract

This paper is devoted to the study of the falsification by fellow-traveller property (FFTP) in Dyer groups. We exhibit a finite generating set for which the associated Cayley graph is a locally finite mediangle graph, and leverage its properties to prove that Dyer groups have the FFTP. It follows that Dyer groups have finitely many cone types, emphasising their role in providing a unified approach to Coxeter groups and graph products of cyclic groups.
Paper Structure (9 sections, 11 theorems, 23 equations, 6 figures)

This paper contains 9 sections, 11 theorems, 23 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Background and preliminary notions
  3. Dyer groups
  4. Mediangle graphs
  5. Cone types
  6. Falsification by fellow-traveller property (FFTP)
  7. Dyer groups have the FFTP
  8. $\mathop{\mathrm{Cay}}\nolimits(D,X_S)$ is a mediangle graph
  9. $(D,X_S)$ has the FFTP

Key Result

Theorem 2.7

Let $D=D(\mathcal{G},m,f)$ be a Dyer group, then $\mathop{\mathrm{Cay}}\nolimits(D,S(X))$ is a mediangle graph.

Figures (6)

  • Figure 1: Example of a Dyer group which is neither a Coxeter group nor a graph product of cyclic groups, for $m>2$.
  • Figure 2: The paths $\gamma$ and $\alpha$$k$-fellow travel.
  • Figure 8: The three path transformations of Definition \ref{['def:transformations']}.
  • Figure 9: The induction step of Lemma \ref{['lemma:hyperplanes-triangles']}. We mark by $\star$ the edges which belong to the hyperplane $\mathcal{H}$.
  • Figure 10: The induction step of Lemma \ref{['lemma:hyperplanes-triangle-convexcycle']}. We mark by $\star$ the edges which belong to the hyperplane $\mathcal{H}$.
  • ...and 1 more figures

Theorems & Definitions (36)

  • Definition 2.1: PaSPaV
  • Example 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Theorem 2.7: Genevois2, corollary of Theorem $2.27$
  • Lemma 2.8
  • proof
  • Remark
  • ...and 26 more