Optimal higher-dimensional Dehn functions for some CAT(0) lattices

Abstract

Let $X=S\times E \times B$ be the metric product of a symmetric space $S$ of noncompact type, a Euclidean space $E$ and a product $B$ of Euclidean buildings. Let $\Gamma$ be a discrete group acting isometrically and cocompactly on $X$. We determine a family of quasi-isometry invariants for such $\Gamma$, namely the $k$-dimensional Dehn functions, which measure the difficulty to fill $k$-spheres by $(k+1)$-balls (for $1\leq k\leq \dim\ X-1$). Since the group $\Gamma$ is quasi-isometric to the associated CAT(0) space $X$, assertions about Dehn functions for $\Gamma$ are equivalent tothe corresponding results on filling functions for $X$. Basic examples of groups $\Gamma$ as above are uniform $S$-arithmetic subgroups of reductive groups defined over global fields. We also discuss a (mostly) conjectural picture for non-uniform $S$-arithmetic groups.