The Twisted Twin of the Grigorchuk Group

Bartholdi, Laurent; Siegenthaler, Olivier

doi:10.1142/s0218196710005728

Cited by 21 publications

(29 citation statements)

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“…We sketch here the computation of the congruence kernel for a new example of group, which is a twisted relative of Grigorchuk's first group [Gri80]. More details will appear in [BS09].…”

Section: Examplesmentioning

confidence: 99%

The congruence subgroup problem for branch groups

2012

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We state and study the congruence subgroup problem for groups acting on rooted tree, and for branch groups in particular. The problem is reduced to the computation of the congruence kernel, which we split into two parts: the branch kernel and the rigid kernel. In the case of regular branch groups, we prove that the first one is Abelian while the second has finite exponent. We also establish some rigidity results concerning these kernels. We work out explicitly known and new examples of non-trivial congruence kernels, describing in each case the group action. The Hanoi tower group receives particular attention due to its surprisingly rich behaviour.Comment: 22 pages, no figur

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“…We sketch here the computation of the congruence kernel for a new example of group, which is a twisted relative of Grigorchuk's first group [Gri80]. More details will appear in [BS09].…”

Section: Examplesmentioning

confidence: 99%

The congruence subgroup problem for branch groups

2012

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“…The proof is presented in section 3.2. It was indicated to us by L.Bartholdi that a shorter proof for Theorem 2 could be given using ideas in [3].…”

Section: Theorem (Grigorchuk [10]) the Relators In The Lysenok Presmentioning

confidence: 99%

Profinite Completion of Grigorchuk's Group Is Not Finitely Presented

Benli

2012

Int. J. Algebra Comput.

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In this paper we prove that the profinite completionĜ of the Grigorchuk group G is not finitely presented as a profinite group. We obtain this result by showing that H 2 (Ĝ, F 2 ) is infinite dimensional. Also several results are proven about the finite quotients G/St G (n) including minimal presentations and Schur Multipliers.In 1980's R.Grigorchuk constructed groups of automorphisms of rooted trees having extraordinary properties. One prototypical example (usually referred as the first Grigorchuk Group and will be denoted by G throughout the paper) was the object of study of several researchers in the last 30 years. It has many interesting properties some of which can be summarized as follows: It is an example of a Burnside group i.e. a finitely generated infinite periodic group. It's growth function has intermediate growth rate and it is the first example of such a group. It is an amenable group which is not elementary amenable. It is not finitely presented but has a special kind of recursive presentation. It is a just infinite branch group i.e. an infinite group whose nontrivial quotients are all finite and its lattice of normal subgroups resembles the binary rooted tree. We refer to [8], [7], [13], [4] for generalities about the Grigorchuk group and related topics. As G is an interesting group from the point view of discrete groups, its profinite completionĜ has several interesting properties in the class of profinite groups. First of all it coincides with the closure of G in the full automorphism group of the binary rooted tree. It is a just-infinite profinite branch group. It has finite width (i.e. the lower central factors have bounded rank). Also it has an interesting universal property that it contains a copy of every 1 countably based pro-2 group. In [2] it was shown thatĜ is a counterexample to a conjecture about just-infinite pro-p groups of finite width.Our main result is about finite presentability ofĜ as a profinite group: Theorem. The profinite completionĜ of G is a 3 generated pro-2 group and is not finitely presented as a profinite group.The main theorem is proved by showing that the cohomology group H 2 (Ĝ, F 2 ) is infinite dimensional. Naturally, to achieve this goal we prove various intermediate results related to finite quotients of G. A step-by-step scheme can be summarized as follows:1. Finding presentations for the finite quotients G n = G/St G (n) (Theorem 1). Using these presentations to compute Schur Multiplier3. Using theorem 2 and the fact that G is a regular branch group showing that H 2 (Ĝ, F 2 ) is infinite dimensional (Theorem 5).Also as byproduct, in theorems 3 and 4, we show that relators of the presentations from Theorem 1 are independent and find minimal presentations for the finite quotients G n .The paper is organized as follows:In section 1 we give basic definitions and properties of self-similar groups and specifically of G. Section 2 is devoted to the discussion of main results. Last section contains the proofs together with intermediate lemmas.Notation:[g, h]...

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“…Then the Dwyer quotients M c (Γ d ) are d-elementary abelian with the following d-ranks. d rk(M c (Γ d )) 3 0 [2] , 1 [3] , 2 [0] , 3 [9] , 4 [1] , 5 [26] , 6 [4] , 7 [77] , 8 [13] , 9 [12] 5 0 [1] , 1 [4] , 2 [2] , 3 [20] , 4 [10] , 5 [100] 6 [2] 7 0 [1] , 1 [2] , 2 [6] , 3 [2] , 4 [14] , 5 [42] , 6 [14] , 7 [34] 11 0 [1] , 1 [2] , 2 [2] , 3 [2] , 4 [10] , 5 [2] , 6 [22] , 7 [22] , 8 [22] , 9 [27] As noted by Laurent Bartholdi and Olivier Siegenthaler, there is a pattern in the ranks of the Dwyer quotients M c (Γ d ). For example, it may holds that for m ∈ N 0 .…”

Section: On the Dwyer Quotients Of Some Fabrykowski-gupta Groupsmentioning

confidence: 99%

“…d M c (Γ d ) (1) [1] (2) [1] (2, 2) [1] (2, 4) [4] (2, 2, 2, 4) [1] 4 (2, 2, 2, 2, 4) [4] (2, 2, 2, 4, 4) [16] (2, 2, 2, 2, 4, 4) [1] (2, 2, 2, 2, 2, 4, 4) [3] (2, 2, 2, 2, 2, 2, 4, 4) [16] (2, 2, 2, 2, 2, 4, 4, 4) [64] (2, 2, 2, 2, 2, 2, 4, 4, 4) [5] (2, 2, 2, 2, 2, 2, 2, 4, 4, 4) [11] (2, 2, 2, 2, 2, 2, 2, 2, 4, 4, 4) [26] (1) [1] (8) [2] (4, 8) [3] (2, 4, 8) [4] (2, 8, 8) [1] (2, 2, 8, 8) [2] (2, 2, 2, 8, 8) [2] (2, 2, 4, 8, 8) [2] (2, 4, 4, 8, 8) [2] (2,4,8,8,8) [2] 8 (2,8,8,8,8) [8] (2,2,8,8,8,8) [4] (2,4,8,8,8,8) [20] (2,…”

Section: On the Dwyer Quotients Of Some Fabrykowski-gupta Groupsmentioning

confidence: 99%