The Dehn function of $\mathrm{SL}(n;\mathbb{Z})$

Abstract

We prove that when $n\ge 5$, the Dehn function of $\mathrm{SL}(n;\mathbb{Z})$ is quadratic. The proof involves decomposing a disc in $\mathrm{SL}(n;\mathbb{R})/\mathrm{SO}(n)$ into triangles of varying sizes. By mapping these triangles into $\mathrm{SL}(n;\mathbb{Z})$ and replacing large elementary matrices by “shortcuts,” we obtain words of a particular form, and we use combinatorial techniques to fill these loops.