On the genus of infinite groups

2011/11/29 by Iain R. Aitchison, Iain Aitchison, Aitchison, Iain +2
Mathematics · #Advanced Operator Algebra Research #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #math.GR #msc:20F65 #msc:57M05

paper · pdf · doi:10.48550/arxiv.1111.7016

arxiv created 2011/11/29 · arxiv updated 2011/12/01

Abstract

We associate to each finite presentation of a group G a compact CW-complex that is a 3-manifold in the complement of 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 `closed group'. A group has genus 0 if and only if it is the fundamental group of a compact orientable 3-manifold.

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