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

TL;DR

This paper investigates the higher-dimensional Dehn functions of certain CAT(0) lattices, providing explicit invariants that measure the complexity of filling spheres in these spaces, with implications for understanding their geometric group properties.

Contribution

The paper determines the optimal higher-dimensional Dehn functions for a class of CAT(0) lattices, linking geometric invariants to algebraic properties of associated groups.

Findings

Explicit formulas for Dehn functions of specific CAT(0) lattices

Connection between Dehn functions and quasi-isometry invariants

Discussion of conjectural behavior for non-uniform S-arithmetic groups

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.