Subgroups of relatively hyperbolic groups of Bredon cohomological dimension 2

Abstract

Abstract A remarkable result of Gersten states that the class of hyperbolic groups of cohomological dimension 2 is closed under taking finitely presented (or more generally FP 2 {\mathrm{FP}_{2}} ) subgroups. We prove the analogous result for relatively hyperbolic groups of Bredon cohomological dimension 2 with respect to the family of parabolic subgroups. A class of groups where our result applies consists of C ′ ⁢ ( 1 / 6 ) {C^{\prime}(1/6)} small cancellation products. The proof relies on an algebraic approach to relative homological Dehn functions, and a characterization of relative hyperbolicity in the framework of finiteness properties over Bredon modules and homological Isoperimetric inequalities.