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 H_{n}, where (Formula presented.). We show that, given any n 2 ∈ N, any group commensurable to H_{n} has the R_{∞} property.

commensurable groups, highly transitive groups, Houghton’s groups, infinite torsion groups, R infinity property, twisted conjugacy, twisted conjugacy classes

Cox, Charles Garnet

522d9ea0-0890-41c6-848a-bcd0a45e2fca

Cox, Charles Garnet

522d9ea0-0890-41c6-848a-bcd0a45e2fca

## 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 H_{n}, where (Formula presented.). We show that, given any n 2 ∈ N, any group commensurable to H_{n} 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

## Catalogue record

Date deposited: 16 Jan 2019 17:30

Last modified: 16 Jan 2019 17:30

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## Contributors

Author:
Charles Garnet Cox

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