{
  "id": "2309.02727",
  "version": "v1",
  "published": "2023-09-06T05:25:54.000Z",
  "updated": "2023-09-06T05:25:54.000Z",
  "title": "Semisimple groups interpretable in various valued fields",
  "authors": [
    "Yatir Halevi",
    "Assaf Hasson",
    "Ya'acov Peterzil"
  ],
  "comment": "arXiv admin note: text overlap with arXiv:2211.00141",
  "categories": [
    "math.LO",
    "math.GR"
  ],
  "abstract": "We study infinite groups interpretable in power bounded $T$-convex, $V$-minimal or $p$-adically closed fields. We show that if $G$ is an interpretable definably semisimple group (i.e., has no definable infinite normal abelian subgroups) then, up to a finite index subgroup, it is definably isogenous to a group $G_1\\times G_2$, where $G_1$ is a $K$-linear group and $G_2$ is a $\\mathbf{k}$-linear group. The analysis is carried out by studying the interaction of $G$ with four distinguished sorts: the valued field $K$, the residue field $\\mathbf{k}$, the value group $\\Gamma$, and the closed $0$-balls $K/\\mathcal{O}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-09-06T05:25:54.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "semisimple groups interpretable",
      "valued field",
      "linear group",
      "definable infinite normal abelian subgroups",
      "finite index subgroup"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}