Converse to strong Strebel in higher dimensions for cubulated hyperbolic groups

Ascertain whether, for a hyperbolic cube complex X with G = π1(X) and cohomological dimension cd(G) = n > 2, the hypothesis that every subgroup of infinite index in G has cohomological dimension less than n implies that H^n(G, ZG) is finitely generated as an abelian group.

Background

Strebel’s theorem and its strengthened version (Brown’s exercise) assert that if a group G has cohomological dimension n and H^n(G, ZG) is finitely generated, then any infinite-index subgroup has strictly smaller cohomological dimension. The paper’s Theorem A serves as a converse in a dimension-2 setting for cubulated hyperbolic groups.

Example 5.3 shows the naive converse fails in dimension 3 for certain hyperbolic cubulated groups (not PD), motivating a refined converse: whether the finite generation of H^n(G, ZG) follows from the cohomological dimension drop for all infinite-index subgroups.