Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Hyperbolic structures on Houghton groups

Anthony Genevois, Geoffrey Tournier

TL;DR

This work computes the poset of hyperbolic structures for Houghton groups, showing that $H_n$ has exactly $n$ focal hyperbolic structures $\\mathcal{F}_i=[\\mathrm{Fix}(R_i)\\cup T]$ and that each one dominates a unique lineal structure, with $n=2$ yielding a single lineal structure and $n\ge3$ yielding uncountably many lineal structures. The authors introduce partial Houghton subgroups and confining automorphisms to analyze focal actions, proving that the two focal structures for $H_n(2)$ are non-comparable and exhaust focal actions for that subgroup. Extending to higher rank, the paper proves that $H_n$ exhibits precisely the $n$ focal structures and describes a rich, uncountable family of lineal structures, all oriented; furthermore, finite extensions $H_n(G)$ realize exactly $k$ focal structures, where $k$ equals the number of $G$-fixed rays on $\{1,\dots,n\}$. The results provide a complete description of the hyperbolic-structure landscape for Houghton groups and construct examples with a single focal structure, addressing questions about the possible numbers of focal actions. The methods combine confining automorphisms, Busemann morphisms, and coboundedness arguments to connect hyperbolic actions to group-theoretic decompositions and abelianization data. This yields both structural insight into Houghton groups and a versatile framework for building groups with prescribed focal-hyperbolic behavior.

Abstract

Given a group $G$, its poset of hyperbolic structures $\mathcal{H}(G)$ encodes all the possible cobounded actions of $G$ on hyperbolic spaces. In this article, we describe the poset $\mathcal{H}(H_n)$ for every Houghton group $H_n$, $n \geq 2$. In particular, we show that $H_n$ admits exactly $n$ focal hyperbolic structures. As an application, we construct the first example of a group admitting exactly one focal hyperbolic structure, answering a question of Abbott, Balasubramanya, and Osin.

Hyperbolic structures on Houghton groups

TL;DR

This work computes the poset of hyperbolic structures for Houghton groups, showing that has exactly focal hyperbolic structures and that each one dominates a unique lineal structure, with yielding a single lineal structure and yielding uncountably many lineal structures. The authors introduce partial Houghton subgroups and confining automorphisms to analyze focal actions, proving that the two focal structures for are non-comparable and exhaust focal actions for that subgroup. Extending to higher rank, the paper proves that exhibits precisely the focal structures and describes a rich, uncountable family of lineal structures, all oriented; furthermore, finite extensions realize exactly focal structures, where equals the number of -fixed rays on . The results provide a complete description of the hyperbolic-structure landscape for Houghton groups and construct examples with a single focal structure, addressing questions about the possible numbers of focal actions. The methods combine confining automorphisms, Busemann morphisms, and coboundedness arguments to connect hyperbolic actions to group-theoretic decompositions and abelianization data. This yields both structural insight into Houghton groups and a versatile framework for building groups with prescribed focal-hyperbolic behavior.

Abstract

Given a group , its poset of hyperbolic structures encodes all the possible cobounded actions of on hyperbolic spaces. In this article, we describe the poset for every Houghton group , . In particular, we show that admits exactly focal hyperbolic structures. As an application, we construct the first example of a group admitting exactly one focal hyperbolic structure, answering a question of Abbott, Balasubramanya, and Osin.

Paper Structure

This paper contains 15 sections, 24 theorems, 127 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Groups with few focal hyperbolic structures.
  3. Acknowledgements.
  4. Preliminaries
  5. Classification of actions on hyperbolic spaces.
  6. Busemann quasimorphisms.
  7. Houghton groups.
  8. Second partial Houghton groups
  9. Confining subsets
  10. Possible focal actions
  11. Hyperbolic structures
  12. Higher rank Houghton groups
  13. Possible focal actions
  14. Hyperbolic structures
  15. Groups with few focal actions

Key Result

Theorem 1.1

Let $n\geq 2$ be an integer. The Houghton group $H_n$ has exactly $n$ focal hyperbolic structures, namely which are pairwise non-comparable. Each $\mathcal{F}_i$ dominates a single lineal hyperbolic structure, namely $[ \mathrm{ker}(\lambda_i) \cup T].$ If $n=2$, $H_n$ has a single lineal hyperbolic structure; if $n \geq 3$, $H_n$ has uncountably many lineal hyperbolic structure, which are pairwi

Figures (1)

  • Figure 1: Structures of the posets $\mathcal{H}(H_2)$ and $\mathcal{H}(H_n)$ for $n \geq 3$.

Theorems & Definitions (66)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Corollary 2.4
  • proof
  • ...and 56 more