New Uniform Diameter Bounds in Pro-p Groups

2014/10/11 by Bradford, Henry
#Combinatorics (math.CO) #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.1410.3007

Abstract

We give new upper bounds for the diameters of finite groups which do not depend on a choice of generating set. Our method exploits the commutator structure of certain profinite groups, in a fashion analogous to the Solovay-Kitaev procedure from quantum computation. We obtain polylogarithmic upper bounds for the diameters of finite quotients of: groups with an analytic structure over a pro-p domain (with exponent depending on the dimension); Chevalley groups over a pro-p domain (with exponent independent of the dimension) and the Nottingham group of a finite field. We also discuss some consequences of our results for random walks on groups.

Related

• Uniform Diameter Bounds in Branch Groups
• A sharp diameter bound for unipotent groups of classical type over Z/pZ
• On finite groups with polynomial diameter
• Upper Bounds For The Diameter Of A Direct Power Of Solvable Groups
• There is no upper bound for the diameter of the commuting graph of a finite group