Subgroups, hyperbolicity and cohomological dimension for totally disconnected locally compact groups

This article is part of the program studying large-scale geometric properties totally disconnected locally compact groups, TDLC-groups, by analogy with theory for discrete groups. We provide a characterization hyperbolic in terms homological isoperimetric inequalities. used to prove main result article: TDLC-groups rational cohomological dimension $\leq 2$, hyperbolicity inherited compactly presented closed subgroups. As consequence, every subgroup automorphism group $\mathrm{Aut}(X)$ negatively curved finite $2$-dimensional building $X$ TDLC-group, whenever acts finitely many orbits on $X$. Examples where this applies include Bourdon's buildings. revisit construction small cancellation quotients amalgamated free products, and verify that it provides examples $2$ when applied products profinite groups over open raise question whether our can be extended if replaced asymptotic dimension. case sketch an argument TDLC-groups.