Isoperimetric inequalities in Hadamard spaces of asymptotic rank two

Gromov’s isoperimetric gap conjecture for CAT(0) 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 Druţu–L.–Papasoglu–Stadler who established a homotopical inequality for Lipschitz 2-spheres with exponents arbitrarily close to 1. In this talk, I will discuss a new homological inequality of the same type for general integral cycles of dimension at least 2, obtained in joint work with Stadler and Urech.