Groups, Geometry, and Dynamics

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Volume 11, Issue 1, 2017, pp. 291–310
DOI: 10.4171/GGD/397

Published online: 2017-04-20

Finitely presented groups and the Whitehead nightmare

Daniele Ettore Otera[1] and Valentin Poénaru[2]

(1) Vilnius University, Lithuania
(2) Université Paris-Sud 11, France

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.

Keywords: Finitely presented groups, weak geometric simple connectivity, quasi-simple

Otera Daniele Ettore, Poénaru Valentin: Finitely presented groups and the Whitehead nightmare. Groups Geom. Dyn. 11 (2017), 291-310. doi: 10.4171/GGD/397