Finitely presented groups and the Whitehead nightmare

TL;DR

This paper introduces a new topological representation framework for QSF groups, demonstrating that such groups admit well-behaved WGSC representations with specific finiteness and closure properties.

Contribution

It establishes that all QSF groups have WGSC representations that are locally finite, equivariant, and have a closed double point set, advancing understanding of their topological structure.

Findings

QSF groups admit WGSC representations with controlled singularities.

Such representations can be made locally finite and equivariant.

The double point set of these representations is closed.

Abstract

We define a `nice representation' of a finitely presented group G as being a non-degenerate essentially surjective simplicial map f from a `nice' space X into a 3-complex associated to a presentation of G, with a strong control over the singularities of f, and such that X is WGSC (weakly geometrically simply connected), meaning that it admits a filtration by simply connected and compact subcomplexes. In this paper we study such representations for a very large class of groups, namely QSF (quasi-simply filtered) groups, where QSF is a topological tameness condition of groups that is similar, but weaker, than WGSC. In particular, we prove that any QSF group admits a WGSC representation which is locally finite, equivariant and whose double point set is closed.