Let T be a d-regular tree (d ≥ 3) and A = Aut(T) its automorphism group. Let Γ be the group generated by n independent Haar-random elements of A. We show that almost surely, every nontrivial element of Γ has finitely many fixed points on T.
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Miklós Abért, Yair Glasner, Most actions on regular trees are almost free. Groups Geom. Dyn. 3 (2009), no. 2, pp. 199–213DOI 10.4171/GGD/54