{
  "id": "1608.03137",
  "version": "v1",
  "published": "2016-08-10T11:37:30.000Z",
  "updated": "2016-08-10T11:37:30.000Z",
  "title": "On the structure of virtually nilpotent compact $p$-adic analytic groups",
  "authors": [
    "William Woods"
  ],
  "categories": [
    "math.GR",
    "math.RA",
    "math.RT"
  ],
  "abstract": "Let $G$ be a compact $p$-adic analytic group. We recall the well-understood finite radical $\\Delta^+$ and FC-centre $\\Delta$, and introduce a $p$-adic analogue of Roseblade's subgroup $\\mathrm{nio}(G)$, 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 $\\mathbf{FN}_p(G)$, an open characteristic subgroup of $G$ contained in $\\mathrm{nio}(G)$. 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 of finite rank, and use this to study the conjugation action of $\\mathrm{nio}(G)$ on $\\mathbf{FN}_p(G)$. We emerge with a structure theorem for $G$, $$1 \\leq \\Delta^+ \\leq \\Delta \\leq \\mathbf{FN}_p(G) \\leq \\mathrm{nio}(G) \\leq G,$$ 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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-08-10T11:37:30.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "adic analytic group",
      "virtually nilpotent compact",
      "sound open normal subgroup",
      "largest orbitally sound open normal"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}