Results on directly finite group algebras, sofic groups and automata, the glued group, character packing, polynomial pairings and line systems with holes, 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.

• One-sided inverses — Lemma 1.1. If ab = 1 and a has a left inverse, then ba = 1. No hypothesis.
• Finite groups — Lemma 2.1. K[G] is stably finite for every finite group G and every field K. No hypothesis.
• Finite supports — Lemma 2.2. A pair with ab = 1 and ba != 1 lives on the finitely generated subgroup generated by the supports. No hypothesis.
• Range projections — Lemma 3.2. The range projection of an idempotent in matrices over the group von Neumann algebra has the same trace. Uses the trace of the Kervaire course.
• Finitely generated fields of characteristic zero — Lemma 3.3. Every finitely generated field of characteristic zero embeds in the complex numbers. Uses the fundamental theorem of algebra (core course).
• Kaplansky's theorem in characteristic zero — Theorem 3.1. K[G] is stably finite for every group G and every field K of characteristic zero. Uses the trace of the Kervaire course.
• Kaplansky's direct-finiteness conjecture fails (OpenAI) — Theorem 5.1. There are a finite field K of characteristic two, a finitely presented group G with odd torsion and a, b in K[G] with ab = 1 and ba != 1; proved in the last lesson. Uses (LIN) and (PLANE).
• A finitely presented group that is neither sofic nor surjunctive (OpenAI) — Corollary 5.2. The group of Theorem 5.1 is not sofic, and b defines an injective cellular automaton that is not surjective. Uses Theorem 5.1.
• Group-algebra elements as automata — Lemma 2.1. c -> T_c is an injective algebra homomorphism into the linear maps of K^G, with T_c T_d = T_cd for right neighbours. No hypothesis.
• The transfer to automata — Proposition 2.2. If ab = 1 over a finite field, T_b is injective with image the fixed points of T_ba, and it misses delta_h whenever (ba - 1)_h != 0. No hypothesis.
• A finite obstruction to surjectivity — Lemma 4.2. A nonsurjective automaton over a countable group misses a finite pattern. No hypothesis.
• Uniform local injectivity — Lemma 4.3. An injective automaton over a countable group determines x(1) from the image on a fixed finite set. No hypothesis.
• Sofic groups are surjunctive (Gromov, Weiss) — Theorem 5.1. Every injective cellular automaton over a sofic group with a finite alphabet is surjective. No hypothesis.
• Counterexamples over finite fields live on nonsofic groups — Corollary 6.1. If ab = 1 != ba in K[G] with K finite, then G is neither surjunctive nor sofic. No hypothesis.
• The incidence criterion (OpenAI) — Theorem 1.1. A regular incidence system of girth at least 12 with vectors satisfying the pairing identities and m < t <= m l^2 gives a finitely presented group and a, b with ab = 1 != ba; proved in lesson 5. Uses (PLANE).
• The glued group is finitely presented — Lemma 2.2. The relators of length at most three define the glued group. No hypothesis.
• The local groups embed (OpenAI) — Lemma 4.1. If the incidence graph has girth at least 12, every copy of F_l^2 embeds in the glued group. Uses (PLANE).
• Local groups as unions of lines — Lemma 1.2. The sum over K_v is 1 plus the sums of sigma_f - 1 over the lines through v. No hypothesis.
• The rectangular identity — Proposition 2.1. UP = U, PT = T and UT = I_m. No hypothesis.
• The twisted extension contains H — Lemma 3.3. H = N x <z> embeds in the twisted extension L, proved with a permutation representation. No hypothesis.
• Orthogonal character idempotents — Lemma 4.2. Q_alpha Q_beta = delta Q_alpha for linear forms on K_*. No hypothesis.
• Twisted averages — Lemma 4.3. tau_v e P_v tau_v^-1 = e Q_(alpha_v). No hypothesis.
• Packing — Lemma 1.1. J_0 C = eP. No hypothesis.
• A square one-sided inverse — Proposition 2.1. a_mat b_mat = I_m. No hypothesis.
• The reverse defect — Proposition 3.2. b_mat a_mat - I_m != 0, certified by the specialization -kappa X^T X - I_t. No hypothesis.
• Matrix units of a finite group algebra — Lemma 4.2. The p_ij in K[(Z/l)^m x| S_m] are linearly independent matrix units. No hypothesis.
• Two elements with ab = 1 != ba — Proposition 4.4. The scalar outputs a and b in K[L x F]. No hypothesis.
• Finite presentation — Lemma 5.1. G = L x F is finitely presented and contains an element of order l. No hypothesis.
• Interpolation on an index simplex — Lemma 1.1. Evaluation on S_d is an isomorphism from polynomials of total degree at most d. No hypothesis.
• Line sums — Lemma 2.1. A polynomial of total degree at most q - 2 sums to zero over every affine line of Q^n. No hypothesis.
• The pairing supplied by holes (OpenAI) — Proposition 3.1. Interpolation polynomials of the holes give vectors with the pairing identities and m < t <= m l^2. No hypothesis.
• A regular line system with holes and girth 12 (OpenAI) — Theorem 1.1. For large h there are holes and a D-regular family of affine lines avoiding them with incidence girth at least 12; proved in the last lesson. Uses (LIN).
• Direction spaces — Lemma 2.1. Planes W_r and spaces E_j any ten of which form a direct sum. No hypothesis.
• The local trade — Lemma 3.1. Replacing the fan of a hole in a plane by parallel lines removes the hole and keeps every other degree. No hypothesis.
• Compatible holes — Lemma 4.1. Holes with the size bounds, one per plane, bounded occupancy and no exceptional rectangle with four witnesses. Uses (LIN) for the bound on s.
• Replacement lines — Lemma 4.2. Replacement lines avoid the holes, are disjoint within a label and form no exceptional rectangle. No hypothesis.
• A vector in a random plane — Lemma 1.1. A fixed nonzero vector lies in a uniform random plane of E_j with probability (q^2 - 1)/(q^(b-1) - 1). No hypothesis.
• Short polygons — Lemma 3.1. Candidate polygons are rectangles or pentagons of three kinds, and their events have probability p_*^(2g). No hypothesis.
• Event bounds — Lemma 4.1. The event probabilities at each slab variable add up to O(q^-7). No hypothesis.
• The local lemma (special case) — Lemma 5.1. Events determined by at most four independent variables, with sums at most 1/16 at each variable, can be avoided together. No hypothesis.
• Slab partitions — Lemma 1.1. A three-dimensional slab has a line partition with a given axis whose other lines are transverse to the axis direction. No hypothesis.
• The trades — Lemma 2.1. After all trades every point off the holes has degree D, and lines of one label are disjoint. No hypothesis.

Reading order

Nine lessons. The first two prove the classical positive results and the consequences for sofic groups and automata; lessons three to five prove the incidence criterion; lesson six constructs the vectors by interpolation; lessons seven to nine construct the line system and assemble the counterexample. The trace comes from the Kervaire course; Linnik's theorem and standard facts on plane graphs are named with free locators.