TL;DR
This paper proves that for n >= 5, the Dehn function of SL(n;Z) is quadratic, using geometric decompositions and combinatorial methods to analyze loop fillings in the group.
Contribution
It establishes the quadratic Dehn function for SL(n;Z) when n >= 5, extending understanding of the group's geometric properties.
Findings
Dehn function of SL(n;Z) is quadratic for n >= 5
Decomposition of discs in SL(n;R)/SO(n) into triangles
Use of shortcuts to simplify words in SL(n;Z)
Abstract
We prove that when n >= 5, the Dehn function of SL(n;Z) is quadratic. The proof involves decomposing a disc in SL(n;R)/SO(n) into triangles of varying sizes. By mapping these triangles 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.
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