Dehn functions of higher rank arithmetic groups of type $A_n$ in products of simple Lie groups

TL;DR

This paper proves that certain higher rank arithmetic groups of type A_n, when embedded in products of simple Lie groups with multiple noncocompact factors, have polynomially bounded Dehn functions, indicating efficient isoperimetric properties.

Contribution

It establishes polynomial bounds on Dehn functions for higher rank arithmetic groups of type A_n in complex Lie group products, extending understanding of their geometric properties.

Findings

Dehn functions are polynomially bounded for these groups

Results apply to groups with at least two noncocompact factors

Builds on work by Bestvina-Eskin-Wortman and Cornulier-Tessera

Abstract

Suppose $\Gamma$ is an arithmetic group defined over a global field $K$, that the $K$-type of $\Gamma$ is $A_n$ with $n \geq 2$, and that the ambient semisimple group that contains $\Gamma$ as a lattice has at least two noncocompact factors. We use results from Bestvina-Eskin-Wortman and Cornulier-Tessera to show that $\Gamma$ has a polynomially bounded Dehn function.