A sharp diameter bound for unipotent groups of classical type over ℤ/pℤ

Abstract

The unipotent subgroup of a finite group of Lie type over a prime field 𝔽 p comes equipped with a natural set of generators; the properties of the Cayley graph associated to this set of generators have been much studied. In the present paper, we show that the diameter of this Cayley graph is bounded above and below by constant multiples of np + n 2 log p , where n is the rank of the associated Lie group. This generalizes the result of [Ellenberg, A sharp diameter bound for an upper triangular matrix group, Harvard University, 1993], which treated the case of SL n (𝔽 p ).