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THE CANCELLATION NORM AND THE GEOMETRY OF BI-INVARIANT WORD METRICS

Part of: Special aspects of infinite or finite groups

Published online by Cambridge University Press:  21 July 2015

MICHAEL BRANDENBURSKY ,
ŚWIATOSŁAW R. GAL ,
JAREK KĘDRA  and
MICHAŁ MARCINKOWSKI
MICHAEL BRANDENBURSKY
Affiliation:
CRM, University of Montreal, Canada e-mail: [email protected]
ŚWIATOSŁAW R. GAL
Affiliation:
Instytut Matematyczny Uniwersytet Wroclawski, pl. Grunwaldzki 2/4, 50-384 Wroclaw, Poland e-mail: [email protected]
JAREK KĘDRA
Affiliation:
Institute of Pure and Applied Mathematics, University of Aberdeen, Fraser Noble Building, Aberdeen AB24 3UE, Scotland e-mail: [email protected]
MICHAŁ MARCINKOWSKI
Affiliation:
Instytut Matematyczny Uniwersytet Wroclawski, pl. Grunwaldzki 2/4, 50-384 Wroclaw, Poland e-mail: [email protected]
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Abstract

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We study bi-invariant word metrics on groups. We provide an efficient algorithm for computing the bi-invariant word norm on a finitely generated free group and we construct an isometric embedding of a locally compact tree into the bi-invariant Cayley graph of a nonabelian free group. We investigate the geometry of cyclic subgroups. We observe that in many classes of groups, cyclic subgroups are either bounded or detected by homogeneous quasimorphisms. We call this property the bq-dichotomy and we prove it for many classes of groups of geometric origin.

MSC classification

Primary: 20F65: Geometric group theory
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 58 , Issue 1 , January 2016 , pp. 153 - 176
Copyright
Copyright © Glasgow Mathematical Journal Trust 2015 

References

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