Large scale geometry of commutator subgroups

Calegari, Danny; Zhuang, Dongping

doi:10.2140/agt.2008.8.2131

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Large scale geometry of commutator subgroups

Danny Calegari and Dongping Zhuang

Algebraic & Geometric Topology 8 (2008) 2131–2146

DOI: 10.2140/agt.2008.8.2131

Abstract

Let $G$ be a finitely presented group, and $G^{'}$ its commutator subgroup. Let $C$ be the Cayley graph of $G^{'}$ with all commutators in $G$ as generators. Then $C$ is large scale simply connected. Furthermore, if $G$ is a torsion-free nonelementary word-hyperbolic group, $C$ is one-ended. Hence (in this case), the asymptotic dimension of $C$ is at least $2$ .

Keywords

commutator subgroup, large-scale connectedness, commutator length, hyperbolic group

Mathematical Subject Classification 2000

Primary: 20F65, 57M07

References

Publication

Received: 29 July 2008

Revised: 1 October 2008

Accepted: 25 October 2008

Published: 10 November 2008

Authors

Danny Calegari
Department of Mathematics California Institute of Technology Pasadena CA 91125 USA
http://www.its.caltech.edu/~dannyc
Dongping Zhuang
California Institute of Technology Department of Mathematics Pasadena, CA 91125 USA

Volume 8, issue 4 (2008)