Group Presentation, p-Adic Analytic Groups and Lattices in SL 2 ( C )

Abstract

groups by using their pro-p completions. This method is based on the observation that if (X; R) is a presentation of the finitely generated (f.g.) group F, then (X; R), considered as a presentation for a pro-p group is actually a presentation for T,;, the pro-p completion of F. First we prove a generalization of a theorem of Golod and Shafarevich (cf. [H]). THEOREM. Let G be a p-adic analytic pro-p group different from Zp (= the p-adic integers). If (X; R) is a minimal presentation of G then 1X;2 IRI? 2