Results on triangle complexes, planar diagrams, hyperbolicity and finite quotients

The course proves the statements below; each link leads to the statement with its assumptions and its proof. The lessons are self-checked by the writing AI; independent mathematical review is not complete.

• A hyperbolic group that is not residually finite — Theorem 1.1. There is a torsion-free word-hyperbolic group that is not residually finite.
• Triangle complex criterion — Theorem 2.1. If every nonempty cyclically reduced closed walk in every vertex link of a connected finite Euclidean triangle complex has angular length at least 2 pi + eta, eta > 0, then closed edge paths whose turns are all at least pi are not nullhomotopic, and the fundamental group is torsion-free and word-hyperbolic.
• Metric preliminaries — Lemma 4.1. The intrinsic metric of a finite Euclidean triangle complex is a compact geodesic metric; local geodesics consist of finitely many straight pieces, and their turns have link distance at least pi.
• Diagonals — Lemma 2.1. Every simple polygon with at least four vertices has a diagonal.
• Cross-cuts — Lemma 2.2. A cross-cut of a simple polygon divides its inside into the insides of the two simple polygons it forms with the two arcs of the boundary.
• Triangulated disks — Theorem 2.4. Every simple polygon is triangulated by diagonals; every triangulation has v - e + t = 1, has an ear if it has two triangles, and the closed inside is piecewise linearly homeomorphic to a convex polygon.
• Euler's formula — Proposition 3.1. A connected plane graph with V vertices, E edges and F faces satisfies V - E + F = 2.
• Models of faces — Proposition 3.3. Every inner face of a connected plane graph inside a convex polygon is an open disk, modelled by a convex polygon whose boundary maps onto the boundary walk of the face.
• Counts of planar diagrams — Lemma 1.2. A planar diagram has v - e + t = 1, its exterior walk has 2e - 3t steps, and every closed edge walk in it is nullhomotopic.
• Edge approximation — Lemma 2.1. A map of a convex polygon to a simplicial complex, along edges on the boundary, is homotopic relative to the marks to a simplicial map; every loop at a vertex is homotopic to an edge loop.
• Reduced fillings — Theorem 2.3. Every nonempty nullhomotopic closed edge path is the exterior walk of a planar diagram mapped to the complex whose interior vertices have cyclically reduced link walks.
• Angular Gauss-Bonnet — Lemma 1.1. In a planar diagram of Euclidean triangles, the sum over the vertices of (2 - n_y) pi - theta_y equals 2 pi.
• Closed paths with large turns — Proposition 2.2. Under the link condition, a nonempty closed edge path whose turns all have link distance at least pi is not nullhomotopic.
• Torsion-freeness — Proposition 3.1. Under the link condition the fundamental group is torsion-free, by shortest loops in free homotopy classes.
• Thin reduced diagrams — Lemma 1.1. Every geodesic triangle in every reduced diagram over the barycentric subdivision is delta-thin, with delta = 10(A + 1) and A = M + 2 pi M / eta.
• Stability of controlled paths — Lemma 2.1. In a geodesic space with delta-thin triangles, a path whose subarcs have length at most lambda times the distance of their endpoints plus C stays within an explicit distance D of a shortest path with the same endpoints, and conversely within 2D.
• Edge paths in diagrams — Lemma 3.1. The edge distance of two vertices of a diagram is at most B_1 times their intrinsic distance.
• Word-hyperbolicity — Proposition 3.2. Under the link condition the fundamental group is word-hyperbolic: geodesic triangles of a Cayley graph are (4H + 2)-thin.
• Punctured lines with few short cycles — Lemma 1.2. Over F_q^20, for large primes q, there are punctured lines whose incidence graph has no cycle of length at most 12, almost every point lying on at least 4r^2 of them, with a numerical budget.
• The links of the complex — Lemma 3.1. The complex satisfies the link condition with eta = 2 pi / 5, and edges with the same label have link distance at least 14 pi / 5 at V and at W.
• Rectangular words — Proposition 4.2. Every rectangular word g_iuv g_iu'v^-1 g_iu'v' g_iuv'^-1 with u != u' and v != v' is nontrivial.
• Rank under reduction — Lemma 1.2. An integer matrix whose rows have norm at most sqrt(r) has rank modulo every prime q > r at least half its real rank.
• The rank obstruction — Theorem 3.4. No homomorphism from the group onto a finite group keeps every rectangular word nontrivial.
• The group is not residually finite — Theorem 4.1. The fundamental group of the complex is torsion-free and word-hyperbolic, and some rectangular word lies in its finite residual.
• Malcev's theorem — Theorem 4.2. A finitely generated group of invertible matrices over a field is residually finite.
• Not linear — Corollary 4.3. Every finite-dimensional linear representation of the group over a field kills its finite residual; the group is not linear.

Reading order

Seven lessons, to be read in order. Lessons 1 to 5 prove the triangle complex criterion; lessons 6 and 7 construct the complex and prove that no finite quotient keeps all rectangular words alive. Every step is proved in the course or in a cited programme lesson.