{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $F$ be a [[field|04C5ZBG1WT]], let $n \\ge 1$, and let $G$ be a [[finitely generated|082DWWXMRJ]] [[subgroup|0X1JXT47GA]] of the [[general linear group|0FCHE902D7]] $\\mathrm{GL}_n(F)$.\nThen exactly one of the following holds:\n\n1. $G$ contains a non-abelian [[free group|0KTW26C80J]].\n2. $G$ is virtually [[solvable|0FXJ7YFMMX]]: it has a solvable subgroup of finite [[index|0DTXBQPA2S]].\n\nIf $F$ has characteristic $0$, the same conclusion holds for every subgroup $G$ of $\\mathrm{GL}_n(F)$, finitely generated or not.\n\n## Remarks\n\nThe two cases exclude each other, because subgroups of virtually solvable groups are virtually solvable and a non-abelian free group is not.\nSuppose $G$ is not virtually solvable.\nTits embeds the relevant field into a suitable local field and finds two elements of $G$ that act on projective space with attracting and repelling points in general position.\nHigh powers of these elements then satisfy the ping-pong lemma, and so generate a free group of rank $2$.\n\nExample: $\\mathrm{SL}_2(\\Z)$ is not virtually solvable, and indeed the matrices $\\begin{pmatrix} 1 \u0026 2 \\\\ 0 \u0026 1 \\end{pmatrix}$ and $\\begin{pmatrix} 1 \u0026 0 \\\\ 2 \u0026 1 \\end{pmatrix}$ generate a free group of rank $2$.\nExample: the group of upper triangular invertible matrices is solvable, so it has no non-abelian free subgroup.\n\nIn positive characteristic, finite generation is needed.\nLet $\\bar{\\F}_p$ be an algebraic closure of $\\F_p$.\nEvery finitely generated subgroup of $\\mathrm{GL}_2(\\bar{\\F}_p)$ is finite, so $\\mathrm{GL}_2(\\bar{\\F}_p)$ has no free subgroup of rank $2$.\nBut it is not virtually solvable, since it has the infinite simple group $\\mathrm{PSL}_2(\\bar{\\F}_p)$ as a quotient of a subgroup.\n\n## Explanations\n\n### L1 human\n\nA group of invertible matrices is either tame or wild.\nTame means that, after passing to a finite-index piece, it is built from commutative layers.\nWild means it contains two matrices that satisfy no relation at all, like the generators of a free group.\nThe theorem says there is nothing in between for finitely generated matrix groups.\n\n### L2 human\n\nThe theorem gives strong consequences for linear groups.\nA finitely generated linear group is [[amenable|CEAT9BESKK]] if and only if it is virtually solvable, since groups with free subgroups are not amenable.\nCombined with the Milnor–Wolf theorem on finitely generated solvable groups, it shows that the [[growth|4CMRM3975W]] of a finitely generated linear group is either polynomial or exponential.\nGroups outside linear groups are said to satisfy the Tits alternative when the same dichotomy holds for their subgroups; examples include [[hyperbolic groups|JWJ8YPR7EB]] and [[Out(F_n)|7Z5ZS7KNXS]].\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "about": [
      "0FCHE902D7",
      "0KTW26C80J",
      "0FXJ7YFMMX"
    ],
    "kind": "statement",
    "msc": [
      "20E05",
      "20G15",
      "20H20"
    ],
    "name": "Tits alternative",
    "role": "theorem",
    "same_as": {
      "wikidata": "Q3613173",
      "wikipedia": "Tits alternative"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Tits_alternative"
      }
    ],
    "tag": "QYXGVES1XM",
    "uses": [
      "04C5ZBG1WT",
      "0FCHE902D7",
      "082DWWXMRJ",
      "0X1JXT47GA",
      "0KTW26C80J",
      "0FXJ7YFMMX",
      "0DTXBQPA2S"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/QYXGVES1XM/",
  "links": [
    {
      "field": "about",
      "target": "0FCHE902D7"
    },
    {
      "field": "about",
      "target": "0KTW26C80J"
    },
    {
      "field": "about",
      "target": "0FXJ7YFMMX"
    },
    {
      "field": "uses",
      "target": "04C5ZBG1WT"
    },
    {
      "field": "uses",
      "target": "0FCHE902D7"
    },
    {
      "field": "uses",
      "target": "082DWWXMRJ"
    },
    {
      "field": "uses",
      "target": "0X1JXT47GA"
    },
    {
      "field": "uses",
      "target": "0KTW26C80J"
    },
    {
      "field": "uses",
      "target": "0FXJ7YFMMX"
    },
    {
      "field": "uses",
      "target": "0DTXBQPA2S"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/QYXGVES1XM.md",
  "msc": [
    "20E05",
    "20G15",
    "20H20"
  ],
  "name": "Tits alternative",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/QYX/QYXGVES1XM.md",
  "status": null,
  "tag": "QYXGVES1XM",
  "uri": "m2://QYXGVES1XM",
  "url": "https://math2.tamnd.com/wiki/tits-alternative/",
  "writing": {
    "tag": "QYXGVES1XM",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}