{
  "id": "1610.05159",
  "version": "v1",
  "published": "2016-10-17T15:21:40.000Z",
  "updated": "2016-10-17T15:21:40.000Z",
  "title": "Character varieties for real forms",
  "authors": [
    "Miguel Acosta"
  ],
  "comment": "20 pages, 4 figures",
  "categories": [
    "math.GT"
  ],
  "abstract": "Let $\\Gamma$ be a finitely generated group and $G$ a real form of $\\mathrm{SL}_n(\\mathbb{C})$. We propose a definition for the $G$-character variety of $\\Gamma$ as a subset of the $\\mathrm{SL}_n(\\mathbb{C})$-character variety of $\\Gamma$. We consider two anti-holomorphic involutions of the $\\mathrm{SL}_n(\\mathbb{C})$ character variety and show that an irreducible representation with character fixed by one of them is conjugate to a representation taking values in a real form of $\\mathrm{SL}_n(\\mathbb{C})$. We study in detail an example: the $\\mathrm{SL}_n(\\mathbb{C})$, $\\mathrm{SU}(2,1)$ and $\\mathrm{SU}(3)$ character varieties of the free product $\\mathbb{Z}/3\\mathbb{Z} * \\mathbb{Z}/3\\mathbb{Z}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-10-17T15:21:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20C15",
      "14L24",
      "14D20"
    ],
    "keywords": [
      "character variety",
      "real form",
      "anti-holomorphic involutions",
      "free product",
      "finitely generated group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}