New uniform diameter bounds in pro- groups
Henry Bradford
Georg-August-Universität Göttingen Germany
![New uniform diameter bounds in pro-$p$ groups cover](/_next/image?url=https%3A%2F%2Fcontent.ems.press%2Fassets%2Fpublic%2Fimages%2Fserial-issues%2Fcover-ggd-volume-12-issue-3.png&w=3840&q=90)
Abstract
We give newupper 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- domain (with exponent depending on the dimension); Chevalley groups over a pro- 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.
Cite this article
Henry Bradford, New uniform diameter bounds in pro- groups. Groups Geom. Dyn. 12 (2018), no. 3, pp. 803–836
DOI 10.4171/GGD/457