Sam Fisher

Free-by-(poly)cyclic groups

In this talk, we will focus on the class of finitely generated residually (polycyclic and virtually nilpotent) (RPVN) groups. While this may seem like an odd condition at first glance, the class of RPVN groups contains many groups familiar to geometric/infinite group theorists, such as all finitely generated RFRS and residually torsion-free nilpotent groups, and consequently all fundamental groups of special cube complexes. The main result we will present is the following: if G is an RPVN group of cohomological dimension 2, then G is free-by-polycyclic if and only if the second L2-Betti number of G vanishes. As a consequence, if G is RPVN, cd(G) = 2, and b_2^(2)(G) = 0, then G is coherent, meaning that all of its finitely generated subgroups are finitely presented. This is joint work with Kevin Klinge.