On the genus of infinite groups

TL;DR

This paper introduces a new topological invariant called genus for finitely presented groups, based on associated 3-manifold complexes, and explores properties of groups with genus 0, linking them to 3-manifold groups.

Contribution

It defines the genus of a group via associated 3-manifold complexes and introduces the concept of closed groups, providing a new perspective on group classification.

Findings

Groups of genus 0 are exactly the fundamental groups of compact orientable 3-manifolds.

The paper constructs a CW-complex for each finite presentation that is a 3-manifold complement.

Examples of groups with various genera are provided.

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.