On the Centralizer Dimension and Lattice of Generalized Baumslag–Solitar Groups

Dudkin, F. A.

doi:10.1134/s0037446618030035

Cited by 4 publications

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“…The subgraphs B c is called the Z-maximal subgraphs. The latter well agreed with Proposition 5 and Remark 6 about Zmaximal subtrees [12].…”

Section: And the Maximal Subtrees Of The Graphssupporting

confidence: 90%

“…This contradicts the hypothesis of the lemma, therefore u ∈ E. Thus, C G (g) ⊆ E and C G (g) = C E (g). Now the conclusion of the lemma follows from Corollary 4 and Corollary 12 [12]. The lemma is proved.…”

Section: And the Maximal Subtrees Of The Graphsmentioning

confidence: 67%

“…Base of induction. If g ∈ E then by corollary 4 [12], the equality Induction step. Suppose that g = a 0 · t ε1 · · · · · t εn · a n , a 0 ∈ E and t is the first stable letter in the reduced form of g. Then we have…”

Section: And the Maximal Subtrees Of The Graphsmentioning

confidence: 99%

“…The subgroup generated by all vertex elements is denoted by E. It coincides with π 1 (T). Given g ∈ E denote by S g (see [12]) the minimal subtree of the tree T such that g ∈ π 1 (S g ).…”

Section: Preliminariesmentioning

confidence: 99%

“…In [12] centralizers of sets of elements and centralizer dimension were described for GBS groups presented by labeled trees. In section 3 we describe the centralizers of elements for all GBS groups with ∆(G) = {1}.…”

Section: Introductionmentioning

confidence: 99%

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Computation of the centralizer dimension of generalized Baumslag–Solitar groups

Dudkin¹

2018

Sib. Elektron. Mat. Izv.

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A finitely generated group G acting on a tree so that all vertex and edge stabilizers are infinite cyclic groups is called a generalized Baumslag-Solitar group (GBS group). The centralizer dimension of a group G is the maximal length of a descending chain of centralizers. In this paper we complete a description of centralizers for unimodular GBS groups. This allows us to find the centralizer dimension of all GBS groups and to establish a way to compute it.Theorem 3 Given a reduced labeled graph A such that π 1 (A) is non-abelian group, ∆(π 1 (A)) = {1} and b 1 (A) = n. Then cdim(π 1 (A)) is odd and 3 cdim(π 1 (A)) 2 · |E(A)| + 1.Moreover, for every odd k, 3 k 2·m+1 there exists a labeled graph B m,n with m edges such that b 1 (B m,n ) = n m, ∆(π 1 (B m,n )) = {1} and cdim(π 1 (B m,n )) = k. Theorem 4 Given a reduced labeled graph A such that π 1 (A) is non-abelian group, ∆(π 1 (A)) = {±1} and b 1 (A) = n. Then cdim(π 1 (A)) is odd and 3 cdim(π 1 (A)) 2 · |E(A)| + 3.Moreover, for every odd k, 3 k 2 · m + 3 there exists a labeled graph B m,n with m edges such that 1 b 1 (B m,n ) = n m, ∆(π 1 (B m,n )) = {±1} and cdim(π 1 (B m,n )) = k.Since the proofs are constructive, we do not just describe the centralizer dimension for GBS groups, but also establish a way to compute it. Remark 5 Given a labeled graph A. There is an algorithm to compute cdim(π 1 (A)).The author is grateful to V. A. Churkin for valuable comments and advice.

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“…The subgraphs B c is called the Z-maximal subgraphs. The latter well agreed with Proposition 5 and Remark 6 about Zmaximal subtrees [12].…”

Section: And the Maximal Subtrees Of The Graphssupporting

confidence: 90%

Section: And the Maximal Subtrees Of The Graphsmentioning

confidence: 67%

Section: And the Maximal Subtrees Of The Graphsmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%