Branch groups

Profinite Groups With Finite Virtual Length

“…E 3.12. There are two inequivalent definitions of a profinite branch group (see, for example, [6] or [2]): one involving a group action on a tree, and another purely algebraic one which includes the previous one as a particular case. According to the latter definition, a profinite branch group is infinite and contains, for each natural number n, an open normal subgroup H n which can be expressed as the direct product of k n copies of a subgroup L n , where {k n } is a strictly increasing sequence of natural numbers.…”

The spectral problem, substitutions and iterated monodromy

“…Another important class of subgroups of Aut X * is the class of branch groups [3,10]. Here we give basic definitions and prove that IM G(z 2 + i) is a regular branch group.…”

Scale-invariant groups

“…For the lamplighter group (for F D Z 2 ) this was first noticed by Grigorchuk andŻuk [GrŻ01], with a simpler proof in [GrNS00], and for general F by [SiSt05]; for the Baumslag-Solitar group by Bartholdi and Šuniḱ [BartŠ06]; for the full affine groups by Brunner and Sidki [BruSi98]. Although it has been well-known that a self-similar group may contain a finite index subgroup that is virtually the direct product of a few copies of itself (the typical examples are branch groups [BartGŠ03], most notably Grigorchuk's groups), it does not seem to have been observed before that some of the not virtually nilpotent examples contain finite index copies of themselves.…”