Results on group algebras, prescribed types, random graphs, bounded patterns, spherical arrangements, cone pictures and torsion-freeness, with their assumptions

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.

• Torsion gives zero divisors — Lemma 1.2. An element of finite order m > 1 gives (1 - g)(1 + g + ... + g^(m-1)) = 0 with both factors nonzero. No hypothesis.
• Unique products exclude zero divisors — Proposition 2.2. If G has the unique-product property, K[G] has no zero divisors for every field K. No hypothesis.
• Zero divisors from two labelled graphs — Proposition 3.2. Two connected immersed labelled graphs with odd intersections of outgoing letter sets, in a group where closed paths have label 1, give alpha beta = 0 in F_2[G], with alpha, beta nonzero when the roots are separated. No hypothesis.
• A torsion-free group algebra with zero divisors (OpenAI) — Theorem 4.1. There are a finitely presented torsion-free group G with a finite two-dimensional classifying space and nonzero alpha, beta in F_2[G] with alpha beta = 0; proved in the last lesson from the named statements (SURF), (SEP), (CELL 2) and (PL 3); the inputs (PI1), (COV), (CELL 1), (PL 1) and (PL 2) are proved in the course on cell complexes, fundamental groups and covering spaces.
• Finite projective planes — Lemma 1.1. Counts of points and lines and their incidences in the projective plane over a finite field. No hypothesis.
• Prescribed types — Lemma 2.1. Every letter lies in the types of np vertices on each side, and the types of a vertex of A and one of B share an odd number of letters. No hypothesis.
• Turn weights — Lemma 3.1. The turn weights are 1/(q+1), p or 1, the proportions on A are at most the weights, and w(t,u) = w(ubar,tbar). No hypothesis.
• The squared-word estimate — Lemma 4.1. The squared word weights of length h >= 3002 add up to at most e^(-h/300). No hypothesis.
• Girth conditioning by switchings — Lemma 2.1. For large n some tuple of matchings has girth at least L = floor(c log n). No hypothesis.
• Prescribed edges — Lemma 3.1. Conditioned on girth at least L, a further prescription holds with probability at most 1/(np - s - r_n). No hypothesis.
• Small diameter — Lemma 4.1. With high probability every component has diameter at most D_0 L. No hypothesis.
• The bounded-pattern estimate — Proposition 1.2. With high probability there is no system of at most K paths, total length between L and CL, with a pairing satisfying the reduction condition by at most I interval pairs and at most eps H unpaired occurrences; eps is chosen before K, C, I. Proved in the sixth lesson. No hypothesis.
• Bounded cycle rank — Lemma 2.1. The image graph of such a system has bounded cycle rank, at most two vertices of degree one, and boundedly many branch vertices. No hypothesis.
• Chains and multiplicities — Lemma 3.1. Paths traverse whole chains, and every edge is traversed at most 2(C + K) times. No hypothesis.
• Stage counts — Lemma 5.1. The edge and vertex counts of the multiplicity stages satisfy sum E_j = H and sum V_j <= H + iota. No hypothesis.
• A common grid — Lemma 1.1. Simultaneous approximation of all comparison shifts by multiples of one block length s. No hypothesis.
• Joins — Lemma 3.1. Every block occurrence takes part in at most one join, and s times the number of unjoined occurrences is at most b + o(L). No hypothesis.
• Link balance — Lemma 4.3. A link component without self-links satisfies 2z <= S + b. No hypothesis.
• Bins — Lemma 5.1. Block words have bins between delta s/2 and a_0 s, and each bin d contains at most e^(2d - delta s) words. No hypothesis.
• Fixed letters — Lemma 2.1. The expected number of realizations of a stage with fixed letters is at most n^(V_j - E_j - k_j) e^(-sum d_i + o(L)). No hypothesis.
• Permitted letters — Lemma 3.1. Count of the letter assignments of a stage through link components and bins. No hypothesis.
• The product over the stages — Lemma 4.1. The product of the stage bounds is at most exp(-(delta/2) H + a_0 b + o(L)). No hypothesis.
• Reduced spherical arrangements — Definition 1.1. Disjoint disks on a sphere whose boundaries carry paths in the graph, with all occurrences paired by disjoint arcs into partners with inverse letters on distinct edges. No hypothesis.
• Short closures — Lemma 2.1. Every path without backtracking closes up to a cyclically immersed path by a path of length at most DL, when the diameters are at most D_0 L. No hypothesis.
• Planar intervals — Lemma 4.1. The pairing of an arrangement cut into N' pieces consists of at most 6N' order-reversing interval pairs inside pieces. Uses (SURF).
• Recursive planar separation — Lemma 1.1. A planar multigraph with m vertices has at most C_sep m / sqrt K vertices whose deletion leaves components of at most K vertices. Uses (SEP).
• Absence of arrangements — Proposition 3.1. With high probability the random graph admits no reduced spherical arrangement. Uses (SURF), (SEP).
• Closed paths have trivial labels — Lemma 1.1. Every closed path in the graph has label 1 in the fundamental group of the cone complex. No hypothesis.
• Cone pictures — Lemma 3.3. Essential spheres and null-homotopies of path labels are represented by cone pictures. Uses (PI1), (COV), (CELL), (PL).
• Band surgery — Lemma 5.1. In a cone picture of least length, partners traverse distinct edges. Uses (SURF), (COV), (CELL).
• Asphericity and root separation — Proposition 6.1. Without reduced spherical arrangements, pi_2 of the cone complex vanishes and paths from the roots to other vertices have nontrivial labels. Uses (SURF), (PI1), (COV), (CELL), (PL).
• Asphericity — Proposition 1.1. All higher homotopy groups vanish and the universal cover is contractible: a finite two-dimensional classifying space. Uses (COV), (CELL).
• The comparison theorem — Lemma 2.1. Chain maps from a free augmented complex to an exact one exist over the identity and are unique up to chain homotopy. No hypothesis.
• The periodic resolution — Lemma 3.1. The periodic resolution of a cyclic group of prime order is exact, and its cohomology with coefficients F_l is F_l in every degree. No hypothesis.
• Torsion-freeness — Theorem 4.1. The group of the construction is torsion-free. Uses (COV), (CELL).
• Finite presentation — Lemma 5.1. The group is generated by the letters with the labels of finitely many closed paths as defining relators. Uses (PI1).

Reading order

Ten lessons. The first proves the cancellation mechanism and states the theorem; the second and third construct the prescribed types and the random graphs; the fourth, fifth and sixth prove the bounded-pattern estimate; the seventh and eighth exclude reduced spherical arrangements; the ninth proves asphericity and root separation by cone pictures; the tenth proves torsion-freeness and assembles the counterexample. Six standard results of topology and graph theory are used as named statements with free locators: surfaces in the sphere (SURF), the planar separator theorem (SEP), fundamental groups of 2-complexes (PI1), covering spaces (COV), cellular homology (CELL) and simplicial approximation (PL); the Hurewicz and Whitehead theorems and cellular homology of finite complexes come from earlier programme lessons.