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# 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

#### 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