Let $G$ be a simple connected Lie group and let $T$ be a $k$-regular tree. Let $\Gamma$ be a lattice in $G\times\textrm{Aut}(T)$.
Assume that the intersection of $\Gamma$ with each direct factor is infinite.
If the projection of $\Gamma$ to $G$ is dense, can we say anything about the projection to $\textrm{Aut}(T)$?