A polynomial isoperimetric inequality for SL(n,Z)

TL;DR

This paper proves that for n≥5, the Dehn function of SL(n,Z) is at most quartic by decomposing loops in the symmetric space and using combinatorial filling techniques.

Contribution

It establishes a polynomial isoperimetric inequality for SL(n,Z) with n≥5, advancing understanding of its geometric group properties.

Findings

Dehn function of SL(n,Z) is at most quartic for n≥5

Decomposition of loops into generalized Siegel sets facilitates the proof

Use of shortcuts and combinatorial techniques to fill loops

Abstract

We prove that when n>=5, the Dehn function of SL(n,Z) is at most quartic. The proof involves decomposing a disc in SL(n,R)/SO(n) into a quadratic number of loops in generalized Siegel sets. By mapping these loops into SL(n,Z) and replacing large elementary matrices by "shortcuts," we obtain words of a particular form, and we use combinatorial techniques to fill these loops.