Let $G$ be a locally compact Hausdorff topological group.

DefinitionA closed normal subgroup $H \unlhd G$ is calledcocompactif $G/H$ is compact with respect to the quotient topology.

Note that a closed normal subgroup $H \unlhd G$ is cocompact if and only if there exist a compact subset $K \subseteq G$ such that $KH = G$ (see this thread for a proof).

My question is:

Question:Does the intersection of two cocompact closed normal subgroup of $G$ have to be cocompact as well?

convex-cocompactsubgroup in the isometry group of Hadamard manifolds $H$ of curvature $\le k<0$. These are discrete isometry groups which are required to act cocompactly on certain convex subsets of $H$. Then intersection of two convex-cocompact subgroups is again convex cocompact. The same works for more general Gromov-hyperbolic spaces, where one uses quasiconvex subsets instead of convex ones. – Misha Nov 5 '12 at 13:11normalsubgroups (and btw not necessarily discrete, but forget this). If $G$ is a locally compact hyperbolic group and $N$ is both closed, normal, and convex cocompact, then it is either compact or cocompact. Well, you don't need to introduce convex cocompact subgroups to show that the intersection of two cocompact closed normal subgroups in a non-amenable LC hyperbolic group is either cocompact or compact. [Btw, in an amenable LC hyperbolic group, the intersection of 2 closed cocompact subgroups can be not convex cocompact.] – YCor Nov 5 '12 at 14:11