Finitely presented groups and the Whitehead nightmare

Abstract

We define a "nice representation" of a finitely presented group $\Gamma$ as being a non-degenerate essentially surjective simplicial map $f$ from a „nice" space $X$ into a 3-complex associated to a presentation of $\Gamma$, 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 to, 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.