Abstract commensurability and the Gupta–Sidki group

Abstract

commensurability and the Gupta-Sidki group Alejandra Garrido AbstractWe study the subgroup structure of the infinite torsion p-groups defined by Gupta and Sidki in 1983. In particular, following results of Grigorchuk and Wilson for the first Grigorchuk group, we show that all infinite finitely generated subgroups of the Gupta-Sidki 3-group G are abstractly commensurable with G or G × G. As a consequence, we show that G is subgroup separable and from this it follows that its membership problem is solvable.Along the way, we obtain a characterization of finite subgroups of G and establish an analogue for the Grigorchuk group.

On conjugacy separability of graph products of groups

“…This means that if G is a group such that every cyclic subgroup of G is p-separable, then G must be a p-group. In general, groups with all subgroups being p-separable are extremely rare -as far as the authors are aware, the only examples are the Grigorchuk's 2-group in the case of pro-2 topology (see [15,Theorem 2]) and the Gupta-Sidki 3-group in the case of pro-3 topology (see [13,Theorem 2]). However, cyclic subgroups of a p-group are finite; hence, being residually-p implies that all cyclic subgroups are separable in the pro-p topology.…”

On the Cantor–Bendixson rank of the Grigorchuk group and the Gupta–Sidki 3 group

“…Since in a branch group rist G (n) ≤ st G (n) and rist G (n) has finite index in G for all n, a branch group has the CSP if and only if every subgroup of finite index contains a rigid stabilizer and every rigid stabilizer contains a level stabilizer. Many of the most studied branch groups have the congruence subgroup property including the Grigorchuk group [5] and the Gupta-Sidki groups [2], [4]; these will be discussed in more detail later.…”

“…Our proof will rely on the following "subgroup induction" theorem due to Grigorchuk-Wilson, for the Grigorchuk group, and Garrido, for the Gupta-Sidki 3-group. Theorem 6.1 ([6, Theorem 3], [4,Theorem 6]). Let G be either the Grigorchuk group or the Gupta-Sidki 3-group.…”

New uniform diameter bounds in pro-$p$ groups

“…Note that K ≤ Stab Γ (p) (1). Let x 1 = [a, b] ∈ K, and for 1 ≤ i ≤ p − 2, define x i+1 = [a, x i ] ∈ K. The following computations were made in [15], generalising results from [27] for Γ (3) . (1).…”