### 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(n*^{c+1}).
It was known that free nilpotent groups of class *c* have
*D(n) = Theta(n*^{c+1}) (i.e. bounded above and below by
contant multiples of *n*^{c+1}). Until recently, the best
proven upper bound was *O(n*^{2c}) 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(n*^{2c}), 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.