A hyperbolic group that is not residually finite
A torsion-free word-hyperbolic group that is not residually finite: the fundamental group of a finite complex of Euclidean triangles with three vertices, built from punctured lines over a finite field, in which one fixed nonidentity element dies in every finite quotient and in every finite-dimensional linear representation. The proof: a criterion for complexes whose vertex links have no short closed walks, through reduced planar fillings, angular Gauss-Bonnet, thin diagrams and a stability lemma, giving torsion-freeness, hyperbolicity and nontrivial closed paths; then a comparison of ranks over the reals and over a fixed prime field, through arrays of permutation matrices, local factorizations and a polynomial certificate over the lines. The course is an exposition of OpenAI's proof (2026).
Seven lessons with proofs, examples and exercises with solutions. Written and self-checked by Claude Opus 5.5 (Anthropic); independent review is not complete.
- Triangle complexes and their links
- Polygons and plane graphs
- Reduced planar fillings
- Angular curvature, closed paths and torsion
- Thin diagrams and hyperbolicity
- Punctured lines and the complex
- The rank obstruction
Theorems and assumptions · Reader and editable sources · Course record
Prerequisites
- The Nullstellensatz and Jacobson rings (Commutative algebra for geometry)
- Polygonal regions and the directions on a circle (Constant-coefficient equations and solvability)
- Fundamental groups and homotopy of loops; compact metric spaces, uniform continuity and Lebesgue numbers (Lebl, Basic Analysis I, Lemma 7.4.10 and Theorem 7.5.11); linear algebra, finite groups and finite fields.