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

Definition A closed normal subgroup $H \unlhd G$ is called cocompact if $G/H$ 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?

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The answer is no in almost the simplest example, namely $G=\mathbf{R}$. – YCor Nov 5 '12 at 10:46
At which examples did you look? – Marc Palm Nov 5 '12 at 12:23
@Mrc: I don't know your the question is addressed to me or Sebastian. Well, the answer to the question is yes in the case of compact groups (trivial) and discrete groups (finite index is stable by finite intersections). Well, $\mathbf{R}$ is one of the simplest examples of LC group neither discrete nor compact. Another maybe even simpler example is $\mathbf{Z}\times\mathbf{R}/\mathbf{Z}$ (which is the direct product of a discrete and a compact group) and the answer to the question is also no in this case. – YCor Nov 5 '12 at 12:42
Yves is right, of course. Nevertheless, there is a setting in which a version of Sebastian's question has positive answer. The concept of a cocompact subgroup generalizes to the one of convex-cocompact subgroup 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 13:11
@Misha: note that Sebastian's setting is normal subgroups (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

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