This paper is part of the program of studying large-scale geometric properties of totally disconnected locally compact groups, TDLC-groups, by analogy with the theory for discrete groups. We provide a characterization of hyperbolic TDLC-groups, in terms of homological isoperimetric inequalities. This characterization is used to prove the main result of this paper: for hyperbolic TDLC-groups with rational discrete cohomological dimension $\le 2$, hyperbolicity is inherited by compactly presented closed subgroups. As a consequence, every compactly presented closed subgroup of the automorphism group $Aut(X)$ of a negatively curved locally finite $2$-dimensional building $X$ is a hyperbolic TDLC-group, whenever $Aut(X)$ acts with finitely many orbits on $X$. Examples where this result applies include hyperbolic Bourdon’s buildings.

We revisit the construction of small cancellation quotients of amalgamated free products, and verify that it provides examples of hyperbolic TDLC-groups of rational discrete cohomological dimension $2$ when applied to amalgamated products of profinite groups over open subgroups.

We raise the question of whether our main result can be extended to locally compact hyperbolic groups if rational discrete cohomological dimension is replaced by asymptotic dimension. We prove that this is the case for discrete groups and sketch an argument for TDLC-groups.

这篇论文是通过类比离散群的理论研究完全不连通的局部紧群 TDLC 群的大尺度几何性质的计划的一部分。我们根据同源等周不等式提供了双曲 TDLC 群的特征。该表征用于证明本文的主要结果：对于具有有理离散上同调维数的双曲 TDLC 群$\le \mathrm{2个}$，双曲性由紧凑呈现的封闭子群继承。因此，自同构群的每个紧呈现的闭子群$奥特(X)$负弯曲的局部有限$\mathrm{2个}$立体建筑$X$是一个双曲 TDLC 群，每当$奥特(X)$作用于有限多条轨道$X$. 适用此结果的示例包括双曲线布尔登建筑。

我们重新审视合并自由乘积的小抵消商的构造，并验证它提供了有理离散上同调维数的双曲 TDLC 群的例子$\mathrm{2个}$当应用于开子群上的有限群的合并产品时。

我们提出了一个问题，如果用渐近维数代替有理离散上同调维数，我们的主要结果是否可以扩展到局部紧双曲群。我们证明这是离散组的情况，并勾勒出 TDLC 组的论点。