Deficiency, relation gap and two-dimensional groups

Abstract

Let [Formula: see text] be a finitely presented, residually finite group and let [Formula: see text] denote the deficiency of [Formula: see text]. Assume that every subgroup [Formula: see text] of finite index in [Formula: see text] satisfies [Formula: see text]. We conjecture that [Formula: see text] has a two-dimensional finite classifying space [Formula: see text]. This conjecture is motivated by an open question about the deficiency gradient of groups and their [Formula: see text]-Betti numbers. In this note, we relate this conjecture to the relation gap problem for group presentations. We verify the pro-[Formula: see text] version of the conjecture, as well as its higher dimensional abstract analogs.