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Simple Expansion Sets and Non-Positive Curvature

Daniel Farley

TL;DR

The paper introduces simple expansion sets as a streamlined framework to construct classifying spaces for generalized Thompson groups and proves that the associated full-support cubical complexes $\Delta^{f}_{\mathcal{B}}$ are non-positively curved, often CAT(0). Under mild hypotheses, these complexes are contractible and provide explicit CAT(0) actions for groups such as $F$, $T$, $V$, and $H_{n}$, as well as for finite similarity structure groups, with finite stabilizers and, in many cases, local finiteness. The approach yields transparent, direct proofs that these groups act geometrically on CAT(0) cubical complexes, complementing and extending prior constructions. Overall, the results offer a concrete combinatorial mechanism to obtain CAT(0) actions and, consequently, to study geometric and algorithmic properties of broad families of groups arising from generalized Thompson-type definitions.

Abstract

An expansion set is a set $\mathcal{B}$ such that each $b \in \mathcal{B}$ is equipped with a set of expansions $\mathcal{E}(b)$. The theory of expansion sets offers a systematic approach to the construction of classifying spaces for generalized Thompson groups. We say that $\mathcal{B}$ is simple if proper expansions are unique when they exist. We will prove that any given simple expansion set determines a cubical complex with a metric of non-positive curvature. In many cases, the cubical complex will be CAT(0). We are thus able to recover proofs that Thompsons groups $F$, $T$, and $V$, Houghton's groups $H_{n}$, and groups defined by finite similarity structures all act on CAT(0) cubical complexes. We further state a sufficient condition for the cubical complex to be locally finite, and show that the latter condition is satisfied in the cases of $F$, $T$, $V$, and $H_{n}$.

Simple Expansion Sets and Non-Positive Curvature

TL;DR

The paper introduces simple expansion sets as a streamlined framework to construct classifying spaces for generalized Thompson groups and proves that the associated full-support cubical complexes are non-positively curved, often CAT(0). Under mild hypotheses, these complexes are contractible and provide explicit CAT(0) actions for groups such as , , , and , as well as for finite similarity structure groups, with finite stabilizers and, in many cases, local finiteness. The approach yields transparent, direct proofs that these groups act geometrically on CAT(0) cubical complexes, complementing and extending prior constructions. Overall, the results offer a concrete combinatorial mechanism to obtain CAT(0) actions and, consequently, to study geometric and algorithmic properties of broad families of groups arising from generalized Thompson-type definitions.

Abstract

An expansion set is a set such that each is equipped with a set of expansions . The theory of expansion sets offers a systematic approach to the construction of classifying spaces for generalized Thompson groups. We say that is simple if proper expansions are unique when they exist. We will prove that any given simple expansion set determines a cubical complex with a metric of non-positive curvature. In many cases, the cubical complex will be CAT(0). We are thus able to recover proofs that Thompsons groups , , and , Houghton's groups , and groups defined by finite similarity structures all act on CAT(0) cubical complexes. We further state a sufficient condition for the cubical complex to be locally finite, and show that the latter condition is satisfied in the cases of , , , and .

Paper Structure

This paper contains 12 sections, 8 theorems, 33 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Simple Expansion Sets
  3. The definition of simple expansion set
  4. The complex $\Delta^{f}_{\mathcal{B}}$
  5. Non-positively curved cubical complexes
  6. The cubical complex condition
  7. The link condition
  8. A sufficient condition for CAT(0)
  9. Applications
  10. Thompson's group $V$ (conclusion)
  11. Finite similarity structure groups
  12. Houghton's groups $H_{n}$

Key Result

Theorem 2.10

(Description of $st_{\uparrow}(v)$; cf. Proposition 2.16 and Theorem 2.22 from Farley) Let $v = \{ b_{1}, \ldots, b_{k} \} \in \left( \Delta^{f}_{\mathcal{B}} \right)^{0}$. The ascending star $st_{\uparrow}(v)$ is abstractly isomorphic to $\mathcal{E}(b_{1}) \times \ldots \times \mathcal{E}(b_{k})$ ∎

Figures (1)

  • Figure 1: Here a cube in $\Delta^{f}_{\mathcal{B}}$ is pictured, where $\mathcal{B}$ is the expansion set for Thompson's group $V$.

Theorems & Definitions (33)

  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Example 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • Theorem 2.10
  • ...and 23 more