We show that if a group G acting faithfully on a rooted tree T has a free subgroup, then either there exists a point w of the boundary ∂T and a free subgroup of G with trivial stabilizer of w, or there exists w ∈ ∂T and a free subgroup of G fixing w and acting faithfully on arbitrarily small neighborhoods of w. This can be used to prove the absence of free subgroups for different known classes of groups. For instance, we prove that iterated monodromy groups of expanding coverings have no free subgroups and give another proof of a theorem by S. Sidki.
Cite this article
Volodymyr V. Nekrashevych, Free subgroups in groups acting on rooted trees. Groups Geom. Dyn. 4 (2010), no. 4, pp. 847–862DOI 10.4171/GGD/110