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Algebra Seminar
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Derek Holt
University of Warwick

The Dehn function of Nilpotent Groups.

Friday 29th, November 12:05-12:55pm, Carslaw 373.

Let G =<X | R> be a group presentation with X and R finite. Then any word w in the group generators that maps to the identity element of G can be expressed, in the free group on X, as a product of conjugates of elements of R. Let w(r) be the least number of such conjugates occurring in such an expression for w and, for a non-negative integer n, let D(n) be the largest value of w(r) over all words w of length n that map onto the identity of G. Then D is called the Dehn function of the presentation. It is not hard to show that different finite presentations of the same group give rise to Dehn functions that differ only in multiplicative constants.

Around 1990, Gromov conjectured that the Dehn function of a finitely generated nilpotent group of class c is O(nc+1). It was known that free nilpotent groups of class c have D(n) = Theta(nc+1) (i.e. bounded above and below by contant multiples of nc+1). Until recently, the best proven upper bound was O(n2c) by C. Hidber. In fact, the number of steps required to reduce w to the identity using the standard commutator collection process can be seen to be bounded above by O(n2c), which provides an alternative (although essentially equivalent) approach to Hidber's bound.

Gromov's conjecture was recently proved by Gersten, Holt and Riley. The trick is to perform commutator collection, but to keep the resulting word in a compressed form, so that its length always remains bounded by a constant multiple of the original length of w. This will be the subject of the talk.