Lipschitz connectivity and filling invariants in solvable groups and buildings

Abstract

Filling invariants of a group or space are quantitative versions of finiteness properties which measure the difficulty of filling a sphere in a space with a ball. Filling spheres is easy in nonpositively curved spaces, but it can be much harder in subsets of nonpositively curved spaces, such as certain solvable groups and lattices in semisimple groups. In this paper, we give some new methods for bounding filling invariants of such subspaces based on Lipschitz extension theorems. We apply our methods to find sharp bounds on higher-order Dehn functions of ${Sol}_{2n+1}$, horospheres in euclidean buildings, Hilbert modular groups and certain $S$–arithmetic groups.