Sigma theory and twisted conjugacy classes

Gonçalves, Daciberg Lima; Kochloukova, Dessislava H.

doi:10.2140/pjm.2010.247.335

Cited by 27 publications

(55 citation statements)

References 42 publications

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“…In §5 we consider the R ∞ -property finite direct product of groups and obtain a strengthening of a result of Gonçalves and Kochloukova [14]. This is a sequel to the paper [14] by Gonçalves and Kochloukova. We reassure the reader that this paper can be read independently of it.…”

Section: Introductionmentioning

confidence: 83%

“…This happens, for example, if Σ c (Γ) is a nonempty finite set. If Σ c (Γ) contains a discrete character class [χ], that is, a class represented by an character χ whose image χ(Γ) ⊂ R is infinite cyclic, then it was observed by Gonçalves and Kochloukova [14] that the character χ itself is fixed by the action of K on Hom(Γ, R). That is, χ • φ = χ for all φ ∈ K ⊂ Aut(Γ).…”

Section: 2mentioning

confidence: 99%

“…It was shown by Gonçalves and Kochloukova [14] that R(φ) = ∞ for all φ in a finite index subgroup of the group of all automorphisms of Γ where Γ = H n , G n . Our main result is the following theorem.…”

Section: Introductionmentioning

confidence: 99%

“…We give two proofs for the case of Houghton groups, neither of which use Σ-theory. However we still need to use the results of [14] in the case of G n . Theorem 1.1.…”

Section: Introductionmentioning

confidence: 99%

“…We reassure the reader that this paper can be read independently of it. Although results from [14] are used, we develop our own proof techniques to go forward.…”

Section: Introductionmentioning

confidence: 99%

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Sigma theory and twisted conjugacy, II: Houghton groups and pure symmetric automorphism groups

Gonçalves

Sankaran

2016

Pacific J. Math.

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Let φ : Γ → Γ be an automorphism of a group Γ. We say that x, y ∈ Γ are in the same φ-twisted conjugacy class and write x ∼ φ y if there exists an element γ ∈ Γ such that y = γxφ(γ −1 ). This is an equivalence relation on Γ called the φ-twisted conjugacy. Let R(φ) denote the number of φ-twisted conjugacy classes in Γ. If R(φ) is infinite for all φ ∈ Aut(Γ), we say that Γ has the R ∞ -property.The purpose of this note is to show that the symmetric group S ∞ , the Houghton groups and the pure symmetric automorphism groups have the R ∞ -property. We show, also, that the Richard Thompson group T has the R ∞ -property. We obtain a general result establishing the R ∞ -property of finite direct product of finitely generated groups. This is a sequel to an earlier work by Gonçalves and Kochloukova, in which it was shown, using the sigma-theory due to Bieri-NeumannStrebel, that for most of the groups Γ considered here, R(φ) = ∞ where φ varies in a finite index subgroup of the automorphisms of Γ.

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Section: Introductionmentioning

confidence: 83%

Section: 2mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…We give two proofs for the case of Houghton groups, neither of which use Σ-theory. However we still need to use the results of [14] in the case of G n . Theorem 1.1.…”

Section: Introductionmentioning

confidence: 99%

“…We reassure the reader that this paper can be read independently of it. Although results from [14] are used, we develop our own proof techniques to go forward.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Sigma theory and twisted conjugacy, II: Houghton groups and pure symmetric automorphism groups

Gonçalves

Sankaran

2016

Pacific J. Math.

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Twisted conjugacy in soluble arithmetic groups

Lins de Araujo,

Santos Rego

2024

Mathematische Nachrichten

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Reidemeister numbers of group automorphisms encode the number of twisted conjugacy classes of groups and might yield information about self‐maps of spaces related to the given objects. Here, we address a question posed by Gonçalves and Wong in the mid‐2000s: we construct an infinite series of compact connected solvmanifolds (that are not nilmanifolds) of strictly increasing dimensions and all of whose self‐homotopy equivalences have vanishing Nielsen number. To this end, we establish a sufficient condition for a prominent (infinite) family of soluble linear groups to have the so‐called property . In particular, we generalize or complement earlier results due to Dekimpe, Gonçalves, Kochloukova, Nasybullov, Taback, Tertooy, Van den Bussche, and Wong, showing that many soluble ‐arithmetic groups have and suggesting a conjecture in this direction.

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On the finite index subgroups of Houghton’s groups

Cox

2022

Arch. Math.

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Houghton’s groups $$H_2, H_3, \ldots $$ H 2 , H 3 , … are certain infinite permutation groups acting on a countably infinite set; they have been studied, among other things, for their finiteness properties. In this note we describe all of the finite index subgroups of each Houghton group, and their isomorphism types. Using the standard notation that d(G) denotes the minimal size of a generating set for G, we then show, for each $$n\in \{2, 3,\ldots \}$$ n ∈ { 2 , 3 , … } and U of finite index in $$H_n$$ H n , that $$d(U)\in \{d(H_n), d(H_n)+1\}$$ d ( U ) ∈ { d ( H n ) , d ( H n ) + 1 } and characterise when each of these cases occurs.

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Sigma theory and twisted conjugacy classes

Cited by 27 publications

References 42 publications

Sigma theory and twisted conjugacy, II: Houghton groups and pure symmetric automorphism groups

Sigma theory and twisted conjugacy, II: Houghton groups and pure symmetric automorphism groups

Twisted conjugacy in soluble arithmetic groups

On the finite index subgroups of Houghton’s groups

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