Isoperimetric inequalities in Hadamard spaces of asymptotic rank two

TL;DR

This paper proves a homological isoperimetric inequality for cycles in Hadamard spaces of asymptotic rank two, advancing understanding of geometric filling inequalities in non-positively curved spaces.

Contribution

It establishes a homological inequality for general cycles in spaces with finite linearly controlled asymptotic dimension, including Hadamard 3-manifolds and CAT(0) cube complexes.

Findings

Proves a homological inequality for cycles in asymptotic rank 2 spaces.

Applies to Hadamard 3-manifolds and finite-dimensional CAT(0) cube complexes.

Abstract

Gromov's isoperimetric gap conjecture for Hadamard spaces states that cycles in dimensions greater than or equal to the asymptotic rank admit linear isoperimetric filling inequalities, as opposed to the inequalities of Euclidean type in lower dimensions. In the case of asymptotic rank 2, recent progress was made by Dru\c{t}u-Lang-Papasoglu-Stadler who established a homotopical inequality for Lipschitz 2-spheres with exponents arbitrarily close to 1. We prove a homological inequality of the same type for general cycles in dimensions at least 2, assuming that the ambient space has finite linearly controlled asymptotic dimension. This holds in particular for all Hadamard 3-manifolds and finite-dimensional CAT(0) cube complexes.