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Let $G$ be a locally compact topological group, $K$ a compact subgroup and $\Gamma$ a discrete subgroup. I try to find a neighbourhood $U$ of the identity such that $\Gamma \cap UK = \Gamma \cap K$. How can I construct such a neighbourhood? If $K$ is trivial, the existence of $U$ follows from the discreteness of $\Gamma$.

Do you have some good references on topological groups?

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    Hewitt and Ross, "Abstract Harmonic Analysis" Volume I, Chapter 2 is a good source for this. They may not answer this exact question, but I believe they present enough for you to answer it yourself.2012-02-22

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