On the wgsc and qsf tameness conditions for finitely presented groups

Abstract

A finitely presented group is weakly geometrically simply connected (wgsc) if it is the fundamental group of some compact polyhedron whose universal covering is wgsc i.e. it has an exhaustion by compact connected and simply connected sub-polyhedra. We show that this condition is almost-equivalent to Brick's qsf property, which amounts to finding an exhaustion approximable by finite simply connected complexes, and also to the tame combability introduced and studied by Mihalik and Tschantz. We further observe that a number of standard constructions in group theory yield qsf groups and analyze specific examples. We show that requiring the exhaustion be made of metric balls in some Cayley complex is a strong constraint, not verified by general qsf groups. In the second part of this paper we give sufficient conditions under which groups which are extensions of finitely presented groups by finitely generated (but infinitely presented) groups are qsf. We prove, in particular, that the finitely presented HNN extension of the Grigorchuk group is qsf.