Most actions on regular trees are almost free

TL;DR

This paper proves that for a group generated by random automorphisms of a regular tree, almost all nontrivial elements have only finitely many fixed points, revealing properties of random subgroups of automorphism groups.

Contribution

It demonstrates that groups generated by Haar-random automorphisms of a regular tree almost surely have nontrivial elements with finitely many fixed points, a new insight into random subgroup behavior.

Findings

Almost surely, nontrivial elements have finitely many fixed points

Randomly generated groups exhibit specific fixed point properties

Results apply to automorphism groups of regular trees

Abstract

Let T be a d-regular tree (d > 2) and A=Aut(T), its automorphism group. Let G be a group generated by n independent Haar-random elements of A. We show that almost surely, every nontrivial element of G has finitely many fixed points on T.