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

2012/05/22 by Leuzinger, Enrico · 2 citations
#Differential Geometry (math.DG) #FOS: Mathematics #Group Theory (math.GR) #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.1205.4923

Abstract

Let X=S× E × 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≤ k≤ 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.

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• Quantitative polynomial cohomology and applications to \textrm L^p-measure equivalence
• Isoperimetric inequalities in Hadamard spaces of asymptotic rank two

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