Hyperbolic actions of Thompson's group $F$ and generalizations

TL;DR

This paper explores the complex structure of hyperbolic actions on Thompson's groups, revealing a rich local structure with uncountably many subposets and contrasting behaviors under certain group extensions.

Contribution

It provides a detailed analysis of hyperbolic structures on Thompson's groups, introducing new concepts like lamplike structures and confining subsets, and shows how these structures behave under group extensions.

Findings

The poset of hyperbolic structures has a simple global shape but a complex local structure.

Uncountably many lamplike subposets exist within the hyperbolic structures.

Adding a semidirect product with Z/2Z collapses the hyperbolic structure complexity.

Abstract

We study the poset of hyperbolic structures on Thompson's group $F$ and its generalizations $F_n$ for $n \geq 2$. The global structure of this poset is as simple as one would expect, with the maximal non-elementary elements being two quasi-parabolic actions corresponding to well-known ascending HNN-extension expressions of $F_n$. However, the local structure turns out to be incredibly rich, in stark contrast with the situation for the $T$ and $V$ counterparts. We show that the subposet of quasi-parabolic hyperbolic structures consists of two isomorphic posets, each of which contains uncountably many subposets of \emph{lamplike} structures, which can be described combinatorially in terms of certain hyperbolic structures on related lamplighter groups. Moreover, each of these subposets, as well as intersections and complements thereof, is very large, in that it contains a copy of the power set of the natural numbers. We also prove that these uncountably many uncountable subposets are not the entire picture, indeed there exists a copy of the power set of the natural numbers consisting entirely of non-lamplike structures. We also prove that this entire vast array of hyperbolic structures on $F_n$ collapses as soon as one takes a natural semidirect product with $\mathbb{Z}/2\mathbb{Z}$. These results are all proved via a detailed analysis of confining subsets, and along the way we establish a number of fundamental results in the theory of confining subsets of groups.