Group laws and free subgroups in topological groups

A proof is given that a permutation group in which different finite sets have different stabilizers cannot satisfy any group law. For locally compact topological groups with this property, almost all finite subsets of the group are shown to generate free subgroups. Consequences of these theorems are derived for: Thompson's group F, weakly branch groups, automorphism groups of regular trees, and profinite groups with alternating composition factors of unbounded degree.