Results on traces of projections, graphical small cancellation, coned Cayley graphs, codes and random subgraph Laplacians, 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.

• The canonical trace on matrices over the reduced group C*-algebra — Lemma 1.1. tau(ab) = tau(ba); tau is positive and faithful on positive elements; tau(I_n) = n. No hypothesis.
• The trace on K_0 — Lemma 2.1. Equivalent projections have equal traces, tau induces a homomorphism tau_*: K_0(C_r*(G)) -> R, and a nonzero projection has a nonzero class. No hypothesis.
• A projection of irrational trace (OpenAI) — Theorem 4.1. There are a finitely generated group G_sur, a prime p > 2, integers k_i with k_(i+1) - k_i -> infinity and a projection b_0 in C_r*(G_sur) of trace sum_i p^(-k_i); proved in lessons 2 to 7. Uses (EXP) and (OSA).
• Sparse sums of powers are irrational — Lemma 4.2. If k_(i+1) - k_i -> infinity, then sum_i p^(-k_i) is irrational, for every integer p >= 2. No hypothesis.
• Failure of the reduced Baum-Connes conjecture (OpenAI) — Corollary 4.3. The class of b_0 is not in the image of the assembly map of G_sur. Uses Theorem 4.1 and (LUCK).
• Walks below the girth — Lemma 1.1. In a simple graph of girth sigma with a reduced labelling, a nonempty closed walk without backtracking has length at least sigma, and the label of a walk without backtracking is freely reduced. No hypothesis.
• Distinct copies — Lemma 2.2. Copies with different indices share no simple path of length at least lambda sigma. Under the standing hypotheses (H1) to (H5) on the labelled graphs.
• Minimal fillings — Lemma 4.3. In a minimal filling no face has a freely trivial label, faces are at least as long as the girth of their copy, and no edge separates two faces or a face and an adjustable side of the same copy. No hypothesis.
• Short arcs and their number — Lemma 5.1. In a minimal filling of a closed walk with m sides, arcs between different copies are shorter than lambda sigma, F <= m - 1 and b <= 6m. Under the standing hypotheses (H1) to (H5) on the labelled graphs.
• Avoidance — Proposition 1.1. A walk of length L in the coned graph between two vertices of a copy P that avoids its center gives d_P <= 12(L+1) lambda sigma(P). Under the standing hypotheses (H1) to (H5) on the labelled graphs.
• Simplicity of the Cayley graph — Proposition 1.2. If lambda < 1/12, the Cayley graph has no loops and no multiple edges. Under the standing hypotheses (H1) to (H5) on the labelled graphs.
• Short fillings — Lemma 2.1. Every closed walk of length n in the coned graph is a product of at most 12n conjugates of closed walks of length at most four. Under the standing hypotheses (H1) to (H5) on the labelled graphs.
• Thin triangles with a universal constant — Proposition 2.2. Geodesic triangles in the coned graph are D_0-thin, with D_0 = 10^6 independent of all the data; thin triangles from chain fillings are Lemma 2.1 of the lesson Thin triangles, overlap balls and the residual forest of the course kadison-kaplansky-projections. Under the standing hypotheses (H1) to (H5) on the labelled graphs.
• Simultaneous overlaps (OpenAI) — Lemma 3.1. Seen from a root R, all vertices that a copy P shares with copies whose centers are no farther from R lie in one intrinsic ball of P of radius floor(12(L_0+1) lambda sigma(P)). Under the standing hypotheses (H1) to (H5) on the labelled graphs.
• Edge expansion from a spectral gap — Lemma 1.1. If dI - Adj >= rho off the constants on a finite connected d-regular graph, then |dW| >= (rho/2) min(|W|, n - |W|). No hypothesis.
• Logarithmic diameter — Lemma 1.2. Edge expansion kappa gives diameter at most 2(log n / log(1 + kappa/d) + 1). No hypothesis.
• Tree balls — Lemma 1.3. In a d-regular graph of girth sigma, a ball of radius r with 2r + 1 < sigma is a tree with 1 + d sum_(j<r) (d-1)^j vertices. No hypothesis.
• The data (OpenAI) — Proposition 2.1. There are labelled 4-regular expanders Theta_i satisfying (H1) to (H5) with a uniform spectral gap, small tree balls and k_i = floor(n_i/2) with k_(i+1) - k_i -> infinity. Uses (EXP) and (OSA).
• Direct sum of copy cycles (OpenAI) — Lemma 3.1. The space of finite cycles of the Cayley graph over F_2 is the direct sum of the cycle spaces of the copies. Uses (EXP) and (OSA).
• Counting subspaces — Lemma 1.1. Among the a-dimensional subspaces of an N-dimensional space over F_p, the proportion containing a given nonzero vector is (p^a - 1)/(p^N - 1). No hypothesis.
• Codes with both distances (OpenAI) — Lemma 1.2. For n_i > 100 there is a code of dimension floor(n_i/2) containing the prototype pattern such that every nonzero vector of the code and of its dual has support larger than n_i/100. No hypothesis.
• Exact relations (OpenAI) — Lemma 2.1. The relations of all copies together impose on a single copy P exactly the dual code C_P^perp. Uses Lemma 3.1 of the lesson on thin triangles and Proposition 2.1 of the lesson on labelled expanders, hence (EXP) and (OSA).
• The group G_sur — Lemma 2.3. V x| Gamma is a countable group generated by (v_*, 1) and the (0, s), s in S, and its nonzero elements of V have order p. No hypothesis.
• Uniform pattern marginals (OpenAI) — Lemma 3.1. Under the product measure on the dual of V, the pattern on a copy of type i is uniformly distributed on its code; it equals the prototype with probability p^(-k_i). Uses Lemma 2.1.
• The Fourier decomposition — Proposition 1.1. The left regular representation of V x| Gamma is unitarily equivalent to the direct integral of the representations pi_omega on l^2(Gamma); norms are suprema over omega and the trace is the average of the diagonal coefficient at 1. No hypothesis.
• The operator in each representation — Lemma 2.2. pi_omega(T) is the Laplacian dI - Adj(G_omega) of the enabled graph, with 0 <= pi_omega(T) <= 2dI. Uses Proposition 1.2 of the lesson on thin triangles.
• Deficit inside an invalid copy (OpenAI) — Lemma 3.1. Every vertex set B of a copy that is not valid satisfies d|B| - 2e(B) >= c|B|. Uses Lemma 1.2 and Lemma 3.1 of the codes lesson and Proposition 2.1 of the lesson on labelled expanders.
• Deficit on the remaining graph (OpenAI) — Lemma 3.2. Every finite set Y of vertices outside the valid copies satisfies d|Y| - 2|E(A)| >= (c/2)|Y| for the enabled edges A inside Y. Uses Lemma 3.1, the cycle-space decomposition and the simultaneous overlap bound, hence (EXP) and (OSA).
• Uniform spectral gap (OpenAI) — Lemma 1.1. For every omega the spectrum of pi_omega(T) lies in {0} together with [epsilon, 2d], epsilon = min(rho, c^2/(8d)), and its kernel is spanned by the indicators of the valid copies. Uses Lemma 3.2 of the Fourier lesson.
• A projection in the reduced algebra (OpenAI) — Proposition 2.1. b_0 = chi(T) is a projection in C_r*(G_sur) whose image in every pi_omega is the kernel projection of pi_omega(T). Uses Lemma 1.1 and the continuous functional calculus.
• The trace (OpenAI) — Proposition 3.1. tau(b_0) = sum_i p^(-k_i). Uses Proposition 2.1, Lemma 3.1 of the codes lesson and the trace formula of the Fourier lesson.

Reading order

Seven lessons. The first sets up traces of projections and K_0, states the Baum-Connes conjecture and Lück's theorem as named statements and deduces the failure from the main theorem; the second to fourth build the group from labelled expanders and prove the geometric estimates for its coned Cayley graph; the fifth builds the module from codes; the sixth and seventh prove the Fourier decomposition, the deficits, the uniform spectral gap and the trace formula. Van Kampen diagrams and plane maps come from the course Van Kampen diagrams; expanders of logarithmic girth and Osajda's labelling theorem are named with free locators; thin triangles from short fillings are proved in the course on the Kadison-Kaplansky conjecture.