On Kac-Moody groups over rings

A split Kac-Moody group $G$ over a topological ring $R$ carries a natural topology defined by Kac and Peterson. In case the underlying topological ring $R$ is $k_\omega$, this turns the Kac-Moody group $G$ into a $k_\omega$ group, which allows for a certain amount of control over this topology. Each $\sigma$-compact locally compact ring is a $k_\omega$ ring, hence the above topology allows to study $S$-arithmetic subgroups of topological Kac-Moody groups over local fields. By using geometric and algebraic methods it is possible to observe Mostow-rigidity of $G(F_q[t^{-1}])$ in $G(F_q((t)))$ for sufficiently large $q$. In my talk I'd like to present the $k_\omega$ topology on $G$ and the Mostow-rigidity of $G(F_q[t^{-1}])$ in $G(F_q((t)))$. However, these methods fail to apply to the arithmetic subgroup $G(Z)$ of a real topological Kac-Moody group $G(\mathbb{R})$. I would be interested to know whether there exists a suitable concept of measure on Kac-Moody groups over local fields that would allow one to use geometric group theory in order to derive Mostow-rigidity uniformly as in the case of semisimple algebraic groups over local fields.