On the structure of virtually nilpotent compact p-adic analytic groups

Abstract

Abstract Let G be a compact p-adic analytic group. We recall the well-understood finite radical Δ + {\Delta^{+}} and FC-centre Δ, and introduce a p-adic analogue of Roseblade’s subgroup nio ⁢ ( G ) {\mathrm{nio}(G)} , the unique largest orbitally sound open normal subgroup of G. Further, when G is nilpotent-by-finite, we introduce the finite-by-(nilpotent p-valuable) radical 𝐅𝐍 p ⁢ ( G ) {\mathbf{FN}_{p}(G)} , an open characteristic subgroup of G contained in nio ⁢ ( G ) {\mathrm{nio}(G)} . By relating the already well-known theory of isolators with Lazard’s notion of p-saturations, we introduce the isolated lower central (resp. isolated derived) series of a nilpotent (resp. soluble) p-valuable group, and use this to study the conjugation action of nio ⁢ ( G ) {\mathrm{nio}(G)} on 𝐅𝐍 p ⁢ ( G ) {\mathbf{FN}_{p}(G)} . We emerge with a structure theorem for G, 1 ≤ Δ + ≤ Δ ≤ 𝐅𝐍 p ⁢ ( G ) ≤ nio ⁢ ( G ) ≤ G , 1\leq\Delta^{+}\leq\Delta\leq\mathbf{FN}_{p}(G)\leq\mathrm{nio}(G)\leq G, in which the various quotients of this series of groups are well understood. This sheds light on the ideal structure of the Iwasawa algebras (i.e. the completed group rings kG) of such groups, and will be used in future work to study the prime ideals of these rings.