Filling functions of arithmetic groups

TL;DR

This paper establishes sharp bounds on the filling functions of arithmetic lattices in higher rank semisimple Lie groups, revealing growth rates that match symmetric spaces in lower dimensions and exponential growth at the rank.

Contribution

It proves new sharp filling inequalities for arithmetic lattices, generalizing previous theorems and confirming longstanding conjectures in geometric group theory.

Findings

Filling volume functions grow at the same rate as symmetric spaces for dimensions less than the rank.

At the rank, filling volume functions grow exponentially.

Confirms conjectures of Thurston, Gromov, and Bux-Wortman.

Abstract

The Dehn function and its higher-dimensional generalizations measure the difficulty of filling a sphere in a space by a ball. In nonpositively curved spaces, one can construct fillings using geodesics, but fillings become more complicated in subsets of nonpositively curved spaces, such as lattices in symmetric spaces. In this paper, we prove sharp filling inequalities for (arithmetic) lattices in higher rank semisimple Lie groups. When $n$ is less than the rank of the associated symmetric space, we show that the $n$-dimensional filling volume function of the lattice grows at the same rate as that of the associated symmetric space, and when $n$ is equal to the rank, we show that the $n$-dimensional filling volume function grows exponentially. This broadly generalizes a theorem of Lubotzky-Mozes-Raghunathan on length distortion in lattices and confirms conjectures of Thurston, Gromov, and Bux-Wortman.