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

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 $Γ$ be a discrete group acting isometrically and cocompactly on $X$. We determine a family of quasi-isometry invariants for such $Γ$, 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 $Γ$ is quasi-isometric to the associated CAT(0) space $X$, assertions about Dehn functions for $Γ$ are equivalent tothe corresponding results on filling functions for $X$. Basic examples of groups $Γ$ 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.