Let $\Gamma=\mathrm{SL}_2(\mathbb{Z}[1/2])$ thought of as a lattice in $G=\mathrm{SL}_2(\mathbb{R})\times\mathrm{SL}_2(\mathbb{Q}_2)$. I want to understand the spherical building for $\Gamma$. Presumably by the usual reduction theory argument this is the spherical building for $G$.
My attempt is that the spherical building for $G$ is the join of the buildings for the factors but I'm struggling to understand what the stabilisers ought to be.
So my question basically is, what are the classes of parabolics of $G$ and $\Gamma$?