Let $G$ be a Lie group, $K\subseteq G$ be a compact group and $N\subseteq$ be a nilpotent group s.t. $N\cap K= \{e\}$. Let $H=N\rtimes K$ be the semidirect product of $N$ and $K$ and let $\Gamma$ be a discrete subgroup of $H$. Is it true that $\Gamma$ has a nilpotent subgroup of finite index. Also, can we guaranty that this index is uniform over all discrete $\Gamma\in H$. In other words, can we find a constant $C$ s.t. for all discrete subgroups $\Gamma$ of $H$, there exists $\tilde{\Gamma}$ a subgroup of $\Gamma$ s.t. $[\Gamma:\tilde{\Gamma}]\leq C$.



Nilpotent subgroups of uniform finite index
Start with this paper: A. I. Mal′cev, On a class of homogeneous spaces, Izvestiya Akad. Nauk. SSSR. Ser. Mat. 13 (1949), 9–32 (look also [here](http://www.math.utoronto.ca/~vtk/NILPnew1.pdf)). This reduces your problem to the case when $\Gamma$ is a lattice in $N\rtimes K$. Next, observe that projection of $\Gamma$ to $K$ has to be a finite group (this should be in M.Raghunathan, "Discrete subgroups of Lie groups"), the result is due to Auslender, you can read his original paper [here](http://www.jstor.org/discover/10.2307/1969945?uid=3739560&uid=2&uid=4&uid=3739256&sid=21101863815891). Now, use [Jordan-Schur Theorem](http://en.wikipedia.org/wiki/Jordan%E2%80%93Schur_theorem): For every compact group $K$ there exists a number $j=j(K)$ so that every finite subgroup of $K$ contains an abelian subgroup of index $\le j$. This is also in Raghunathan's book, see also wikipedia article [here](http://en.wikipedia.org/wiki/Jordan%E2%80%93Schur_theorem) and Tao's blog [here](http://terrytao.wordpress.com/2011/10/05/the-jordan-schur-theorem/). 


