Mathematical Research Letters

Volume 19 (2012)

Number 3

On the genus of infinite groups

Pages: 601 – 612

DOI: http://dx.doi.org/10.4310/MRL.2012.v19.n3.a8

Authors

Iain Aitchison (Department of Mathematics and Statistics, University of Melbourne, Australia)

Lawrence Reeves (Department of Mathematics and Statistics, University of Melbourne, Australia)

Abstract

We associate to each finite presentation of a group $G$ acompact CW-complex that is a 3-manifold in the complementof a point, and whose fundamental group is isomorphic to$G$. We use this complex to define a notion of genus for$G$ and give examples, and also define a notion of ‘closedgroup’. A group has genus 0 if and only if it is thefundamental group of a compact orientable 3-manifold.

Keywords

geometric group theory, genus

2010 Mathematics Subject Classification

20F65, 57M05

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