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A note on the R property for groups FAlt(X)⩽G⩽Sym(X)

A note on the R property for groups FAlt(X)⩽G⩽Sym(X)
A note on the R property for groups FAlt(X)⩽G⩽Sym(X)

Given a set X, the group Sym(X) consists of all bijective maps from X to X, and FSym(X) is the subgroup of maps with finite support i.e. those that move only finitely many points in X. We describe the automorphism structure of groups FSym(X) ≤ G ≤ Sym(X) and use this to state some conditions on G for it to have the R property. Our main results are that if G is infinite, torsion, and FSym(X) ≤ G ≤ Sym(X), then it has the R property. Also, if G is infinite and residually finite, then there is a set X such that G acts faithfully on X and, using this action, 〈G FSym(X)〉 has the R property. Finally we have a result for the Houghton groups, which are a family of groups we denote Hn, where (Formula presented.). We show that, given any n 2 ∈ N, any group commensurable to Hn has the R property.

commensurable groups, highly transitive groups, Houghton’s groups, infinite torsion groups, R infinity property, twisted conjugacy, twisted conjugacy classes
0092-7872
Cox, Charles Garnet
522d9ea0-0890-41c6-848a-bcd0a45e2fca
Cox, Charles Garnet
522d9ea0-0890-41c6-848a-bcd0a45e2fca

Cox, Charles Garnet (2018) A note on the R property for groups FAlt(X)⩽G⩽Sym(X). Communications in Algebra. (doi:10.1080/00927872.2018.1499917).

Record type: Article

Abstract

Given a set X, the group Sym(X) consists of all bijective maps from X to X, and FSym(X) is the subgroup of maps with finite support i.e. those that move only finitely many points in X. We describe the automorphism structure of groups FSym(X) ≤ G ≤ Sym(X) and use this to state some conditions on G for it to have the R property. Our main results are that if G is infinite, torsion, and FSym(X) ≤ G ≤ Sym(X), then it has the R property. Also, if G is infinite and residually finite, then there is a set X such that G acts faithfully on X and, using this action, 〈G FSym(X)〉 has the R property. Finally we have a result for the Houghton groups, which are a family of groups we denote Hn, where (Formula presented.). We show that, given any n 2 ∈ N, any group commensurable to Hn has the R property.

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More information

Accepted/In Press date: 23 June 2018
e-pub ahead of print date: 19 November 2018
Keywords: commensurable groups, highly transitive groups, Houghton’s groups, infinite torsion groups, R infinity property, twisted conjugacy, twisted conjugacy classes

Identifiers

Local EPrints ID: 427454
URI: https://eprints.soton.ac.uk/id/eprint/427454
ISSN: 0092-7872
PURE UUID: d25003c3-32a5-4a5b-b57e-524e4eb896c8

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Date deposited: 16 Jan 2019 17:30
Last modified: 16 Jan 2019 17:30

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Author: Charles Garnet Cox

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