Most actions on regular trees are almost free

Abstract

Let $T$ be a $d$-regular tree ($d\ge 3$) and $A=\mathrm{Aut}(T)$ its automorphism group. Let $\Gamma$ be the group generated by $n$ independent Haar-random elements of $A$. We show that almost surely, every nontrivial element of $\Gamma$ has finitely many fixed points on $T$.