Let $\Gamma$ be a discrete co-compact subgroup of $\textit{SL}_2(\mathbb{C})$ and let $\pi:\mathcal{X}\mapsto B$ be a deformation of the homogeneous space $\textit{SL}_2(\mathbb{C})/\Gamma$. When I compute the complex analogous of the mapping class group of fibers of $\pi$, that is $\textit{Aut}(\pi^{-1}(b))/\textit{Aut}^0(\pi^{-1}(b))$, I find that, for some $b$, it is isomorphic to the quotient of the normalizer $N_{\textit{SL}_2(\mathbb{C})}(\Gamma)$ of $\Gamma$ in $\textit{SL}_2(\mathbb{C})$ by $\Gamma$ itself.
My questions are :
Does $N_{\textit{SL}_2(\mathbb{C})}(\Gamma)/\Gamma$ have infinite number of component ?
Under which hypothesis on $\Gamma$ can we have infinite number of components ?