# Existence of open dense subset in a Lie group

Let $G=S\cdot R$ be a Levi-Malcev decomposition of a connected complex Lie group and $\Gamma$ a discrete subgroup of $G$ such that the subgroups
$\Gamma_g := \Gamma\cap (gSg^{-1})$ are finite for all $g\in G$. Is it true that there exist a dense open subset W of G such that the
$\Gamma_g$ are all isomorphic for all $g\in W$?

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