On conjugacy of GGS-groups

Abstract

Each GGS-group on the p-adic tree is defined by a {{(p-1)}} -tuple of integers modulo p, called the defining tuple. It is proved that two GGS-groups are conjugate (equivalently, by a theorem of Grigorchuk and Wilson, isomorphic) in the automorphism group of the tree if and only if their defining tuples can be transformed into each other by multiplication by an integer modulo p and performing certain reorderings of the tuple. Finally, the number of isomorphism classes of GGS-groups is calculated.

Groups of small period growth

“…We now construct K r as a group of automorphisms of , using a construction much in spirit of the Gupta–Sidki p -groups or the second Grigorchuk group. In fact, K r is a (constant) spinal group in the terminology of [3, 9 ].…”

“…This is a group generated by r + 1 involutions. For r = 2, this group contains elements of infinite order, but for r > 2, the groups K r are periodic by [9, Theorem A]. We do not need to rely on this result since the bounds establishing slow period growth also show that K r is periodic for r > 4.…”

Beauville structures for quotients of generalised GGS-groups

“…Indeed, for a time it was unclear whether such groups even existed, until the first examples were given in [11]. We also want to emphasise that for the GGS-groups acting on the p-adic tree, if one were to consider the case p = 2 there is only one group, which is the infinite dihedral group, hence its quotients do not admit Beauville structures, and for p = 3 only one out of the three isomorphism classes of such groups has quotients admitting Beauville structures; see [15] and [18].…”

Hausdorff dimension of the second Grigorchuk group

“…We recall that the second Grigorchuk group acts on the 4adic tree, with generators a and b, where a cyclically permutes the four maximal subtrees rooted at the first level vertices, whereas b fixes the first-level vertices pointwise and is recursively defined by the tuple (a, 1, a, b) which corresponds to the action of b on the four maximal subtrees. The second Grigorchuk group was generalised by Vovkivsky [25] to the family of Grigorchuk-Gupta-Sidki (GGS-)groups acting on the p n -adic tree, for p any prime and n ∈ N. Although the family of GGS-groups acting on the p-adic tree, for p an odd prime, has been well studied (see for instance [9,10,12,[18][19][20]), the more general GGS-groups, apart from in [25], have only recently been considered in more depth (see [5,6]).…”