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Stable homology of Higman--Thompson groups via scanning methods

Marie-Camille Delarue

TL;DR

The paper provides a new scanning-based proof of homological stability for the Higman–Thompson groups $V_{n,r}$, identifying the stable homology with that of the basepoint component of the infinite loop space of the mod $(n-1)$ Moore spectrum, $\Omega^{\infty}_0 \mathbb{M}_{n-1}$. By leveraging Thumann's operad framework and cube-cutting operads, the authors construct a topological model for the disjoint union of these groups and establish a delooping via scanning on a cobordism-like category of tree embeddings. The key steps include showing the equivalence $B\mathrm{HT}_N \simeq B\mathcal{S}(\mathcal{O}_{1,n})$, developing a semi-simplicial resolution to realize a delooping, and applying group-completion theorems to identify $BV_{n,\infty}$ with $\Omega^{\infty}_0 \mathbb{M}_{n-1}$. The result provides a uniform, geometrically grounded explanation of the stable homology and demonstrates the effectiveness of scanning techniques in high-dimensional group homology. Overall, the work connects algebraic properties of $V_{n,r}$ to stable homotopy-theoretic invariants, with potential extensions to higher-dimensional cube-cutting operads.

Abstract

The Higman--Thompson groups $V_{n,r}$ consist of piecewise linear automorphisms of $r$ intervals where cut points and slopes are $n$-adic. Szymik and Wahl prove homological stability for this family of groups as $r$ increases, and compute the stable homology to be that of the infinite loop space of the Moore spectrum. We give a new proof of this result using scanning methods on a topological model for the disjoint union of these groups. We use Thumann's framework of operad groups to build this model.

Stable homology of Higman--Thompson groups via scanning methods

TL;DR

The paper provides a new scanning-based proof of homological stability for the Higman–Thompson groups , identifying the stable homology with that of the basepoint component of the infinite loop space of the mod Moore spectrum, . By leveraging Thumann's operad framework and cube-cutting operads, the authors construct a topological model for the disjoint union of these groups and establish a delooping via scanning on a cobordism-like category of tree embeddings. The key steps include showing the equivalence , developing a semi-simplicial resolution to realize a delooping, and applying group-completion theorems to identify with . The result provides a uniform, geometrically grounded explanation of the stable homology and demonstrates the effectiveness of scanning techniques in high-dimensional group homology. Overall, the work connects algebraic properties of to stable homotopy-theoretic invariants, with potential extensions to higher-dimensional cube-cutting operads.

Abstract

The Higman--Thompson groups consist of piecewise linear automorphisms of intervals where cut points and slopes are -adic. Szymik and Wahl prove homological stability for this family of groups as increases, and compute the stable homology to be that of the infinite loop space of the Moore spectrum. We give a new proof of this result using scanning methods on a topological model for the disjoint union of these groups. We use Thumann's framework of operad groups to build this model.
Paper Structure (22 sections, 35 theorems, 61 equations, 11 figures)

This paper contains 22 sections, 35 theorems, 61 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Homological stability.
  3. Scanning techniques.
  4. Proof strategy.
  5. Organization of the paper.
  6. Acknowledgements.
  7. Preliminaries
  8. Object change in classifying spaces of categories
  9. Description of the Higman--Thompson groups
  10. Trees and forests.
  11. Paired forest diagrams.
  12. Operad groups
  13. Cancellative calculus of fractions
  14. Cube cutting operads
  15. Embedding spaces of trees
  16. ...and 7 more sections

Key Result

Theorem 1.1

For all $n\geq 2$, there is a homology equivalence :

Figures (11)

  • Figure 1: Illustration of an element in Thompson's group $V$
  • Figure 2: Philosophy of the scanning map
  • Figure 3: Example of a reduction of paired forest diagrams
  • Figure 4: Example of an element in $\mathcal{E}_N(T)$
  • Figure 5: An element in $\mathcal{G}_N(T)$
  • ...and 6 more figures

Theorems & Definitions (87)

  • Theorem 1.1: szymikwahl
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • Definition 2.5
  • Example 2.6
  • Lemma 2.7
  • Proposition 2.8
  • Definition 2.9
  • ...and 77 more