{
  "id": "free-group",
  "domain": "algebra",
  "type": "Concept",
  "title": "Free Group",
  "slug": "free-group",
  "url": "/mathematics/algebra/free-group/",
  "summary": "The most unconstrained group on given generators — no relations beyond group axioms. Universal property. Nielsen-Schreier: subgroups of free groups are free. Word problem undecidable for general presentations. Banach-Tarski connection via F₂ ⊂ SO(3).",
  "added_to_codex": "2026-07-09",
  "last_verified": "2026-07-09",
  "sources": [
    {
      "tier": 1,
      "citation": "Lyndon, R.C. and Schupp, P.E. (2001). Combinatorial Group Theory. Springer."
    },
    {
      "tier": 1,
      "citation": "Serre, J.-P. (1980). Trees. Springer."
    },
    {
      "tier": 2,
      "citation": "Magnus, W., Karrass, A. and Solitar, D. (2004). Combinatorial Group Theory. Dover."
    },
    {
      "tier": 3,
      "citation": "Stillwell, J. (1993). Classical Topology and Combinatorial Group Theory. 2nd ed."
    }
  ],
  "entry_status": "live",
  "reading_levels_available": [
    "curious",
    "exploring",
    "deep_dive"
  ],
  "completeness": 1.0,
  "codex_version": "2.5"
}