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A hyperbolic free-by-cyclic group determined by its finite quotients

Part of: Structure and classification of infinite or finite groups Special aspects of infinite or finite groups

Published online by Cambridge University Press:  04 April 2025

Naomi Andrew Open the ORCID record for Naomi Andrew [Opens in a new window] ,
Paige Hillen ,
Robert Alonzo Lyman Open the ORCID record for Robert Alonzo Lyman [Opens in a new window]  and
Catherine Eva Pfaff Open the ORCID record for Catherine Eva Pfaff [Opens in a new window]
Naomi Andrew
Affiliation:
Mathematical Institute, University of Oxford, Oxford, OX2 6GG, UK
Paige Hillen
Affiliation:
Department of Mathematics, University of California - Santa Barbara, Santa Barbara, 93117, USA
Robert Alonzo Lyman*
Affiliation:
Department of Mathematics, Rutgers University - Newark, Newark, NJ 07102, USA
Catherine Eva Pfaff
Affiliation:
Department of Mathematics & Statistics, Queen’s University, Kingston, ON K7L 3N6, Canada Institute for Advanced Study, Princeton, NJ 08540, USA
*
Corresponding author: Robert Alonzo Lyman; Email: robbie.lyman@rutgers.edu
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Abstract

We show that the group $ \langle a,b,c,t \,:\, a^t=b,b^t=c,c^t=ca^{-1} \rangle$ is profinitely rigid amongst free-by-cyclic groups, providing the first example of a hyperbolic free-by-cyclic group with this property.

Keywords

free-by-cyclic grouphyperbolic groupprofinite rigidityresidual finitenessOut(F_n)

MSC classification

Primary: 20E26: Residual properties and generalizations; residually finite groups
Secondary: 20E05: Free nonabelian groups 20E36: Automorphisms of infinite groups 20F65: Geometric group theory

Information

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 67 , Issue 3 , September 2025 , pp. 500 - 502
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust

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