  In 1996, Gersten proved that finitely presented subgroups of a word
hyperbolic group of integral cohomological dimension 2 are hyperbolic. We use
isoperimetric inequalities over arbitrary rings to extend this result to any
ring. In particular, we study the discrete isoperimetric function and show that
its linearity is equivalent to hyperbolicity, which is also equivalent to it
being subquadratic. We further use these ideas to obtain conditions for
subgroups of higher rank hyperbolic groups to be again higher rank hyperbolic
of the same rank. The appendix discusses the equivalence between isoperimetric
inequalities and coning inequalities in the simplicial setting and the general
setting, leading to combinatorial definitions of higher rank hyperbolicity in
the setting of simplicial complexes and allowing us to give elementary
definitions of higher rank hyperbolic groups.
