The superreflexive fixed-point conjecture for higher rank groups and lattices

Let $G=\prod_{i=1}^{m}\mathbf{G}_{i}(k_{i})$, where each $k_i$ is a local field and each $\mathbf{G}_{i}(k_i)$ is the group of $k_i$-points of a Zariski connected simple $k_i$-algebraic group of $k_i$-rank at least $2$. Let $B$ be a superreflexive topological vector space, meaning a topological vector space isomorphic to a uniformly convex Banach space. Property $({\,\overline{\!F}\!}_{B})$ means that every uniformly equicontinuous affine $G$-action on $B$ has a fixed point, while property $({\,\overline{\!T}\!}_{B})$ means that every uniformly equicontinuous linear representation on $B$ has no almost invariant vectors in the quotient by its invariant vectors. The superreflexive fixed-point conjecture. Higher rank groups $G$ as above and their lattices have property $({\,\overline{\!F}\!}_{B})$, and hence property $({\,\overline{\!T}\!}_{B})$, for all superreflexive $B$. The preceding theorem establishes the analogous assertion for the specified $L^p(\mu)$-related spaces, but the extension to all superreflexive spaces is proposed here and remains unresolved in the supplied text.