The Dehn function of \(\text{SL}(n;\mathbb Z)\). (English) Zbl 1276.20053

Young, Robert

Ann. Math. (2) 177, No. 3, 969-1027 (2013); erratum 178, No. 3, 1199 (2013).

The Dehn function of a group equipped with a generating set is a geometric invariant that measures the difficulty in reducing a word written in the generators that represents the identity to the trivial word. In the paper under review, the author proves that for \(n\geq 5\), the Dehn function of \(\mathrm{SL}(n,\mathbb Z)\) is quadratic. Thurston conjectured that this holds for all \(n\geq 4\). The result for \(n=4\) is still open.
The proof reduces the probem of filling of words in \(\mathrm{SL}(n,\mathbb Z)\) to the problem of filling curves in the symmetric space \(\mathrm{SL}(n,\mathbb R)/\mathrm{SO}(n)\). The author’s proof involves then the decomposition of a disk 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”, the author obtains words of a particular form and he uses combinatorial techniques to fill in these loops. In particular, the arguments involve the question of filling in words in parabolic subgroups of the group.

Reviewer: Athanase Papadopoulos (Strasbourg)

MSC:

20F65	Geometric group theory
57S25	Groups acting on specific manifolds
20F05	Generators, relations, and presentations of groups
20F10	Word problems, other decision problems, connections with logic and automata (group-theoretic aspects)

Keywords:

quadratic Dehn functions; filling inequalities; special linear groups; word problem

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