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.

1. Triangle complexes and their links
2. Polygons and plane graphs
3. Reduced planar fillings
4. Angular curvature, closed paths and torsion
5. Thin diagrams and hyperbolicity
6. Punctured lines and the complex
7. 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.