Finitely presented groups and the Whitehead nightmare

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.

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