A Proof that Thompson's Groups have Infinitely Many Relative Ends

TL;DR

This paper proves that Thompson's groups F, T, and V have infinitely many ends relative to specific subgroups and establishes that T and V possess Serre's property FA, indicating fixed points in their actions on trees.

Contribution

It provides new proofs that Thompson's groups have infinitely many relative ends and confirms T and V have property FA, extending understanding of their geometric properties.

Findings

F, T, V have infinitely many ends relative to certain subgroups

T and V have Serre's property FA, ensuring fixed points in actions on trees

The proof for property FA is based on Ken Brown's notes

Abstract

We show that each of Thompson's groups F, T, and V have infinitely many ends relative to certain subgroups. We go on to show that T and V both have Serre's property FA, i.e., any action of T or V on a tree will have a fixed point. (The proof of the latter statement was originally due to Ken Brown, and our proof is based on his notes.)