# Groups, Geometry, and Dynamics

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**Volume 8, Issue 2, 2014, pp. 441–466**

**DOI: 10.4171/GGD/233**

Published online: 2014-07-08

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

Enrico Leuzinger^{[1]}(1) Karlsruhe Institute of Technology (KIT), Germany

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 to the 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.

*Keywords: *Filling functions, higher-dimensional Dehn functions, isoperimetric inequalities, symmetric spaces, Euclidean buildings, lattices, $S$-arithmetic groups

Leuzinger Enrico: Optimal higher-dimensional Dehn functions for some CAT(0) lattices. *Groups Geom. Dyn.* 8 (2014), 441-466. doi: 10.4171/GGD/233