A condition that prevents groups from acting nontrivially on trees

TL;DR

This paper introduces a simple criterion to determine when groups have Serre's property FA, demonstrated on groups like Aut(F_n) and SL(n,Z) for n≥3, based on finite subgroup patterns.

Contribution

The paper provides a new, straightforward criterion for verifying property FA in groups, applicable to important classes like automorphism groups and special linear groups.

Findings

Aut(F_n) satisfies property FA for n≥3

SL(n,Z) satisfies property FA for n≥3

The criterion is based on finite subgroup patterns

Abstract

We describe a simple criterion for showing that a group has Serre's property FA. By exhibiting a certain pattern of finite subgroups, we show that this criterion is satisfied by Aut(F_n) and SL(n,Z) when n>=3.