Results on coboundary expansion, on coset complexes and on the complexes of the theorem, 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 coboundary does not increase the norm by more than the factor i + 2 — Lemma 2.2. For a cochain f of degree i on a pure complex, the weighted norm of its coboundary is at most i + 2 times the norm of f.
• Coboundary expansion and cohomology — Proposition 3.2. A coboundary expansion inequality in degree i forces the cohomology of degree i with coefficients in the field with two elements to vanish; conversely an inequality relative to the cocycles and the vanishing of cohomology give the inequality relative to the coboundaries.
• The full simplex is a coboundary expander — Proposition 4.3. For the complex of all subsets of a set with s + 1 elements, the norm of the coboundary of a cochain of degree i < s is at least (s + 1)/(s - i) times its distance from the coboundaries.
• Weights on a skeleton — Lemma 5.1. If every face of dimension s of a pure complex of dimension n lies in between a and b faces of dimension n, the weights of the s-skeleton and of the complex differ in every degree up to s by factors between Lambda/b and Lambda/a, with one number Lambda for all degrees.
• Expansion passes to a skeleton — Corollary 5.2. Under the assumption of the lemma on weights, an expansion inequality with the constant eta for the complex gives the inequality with the constant a eta/b for its s-skeleton, in the degrees below s.
• Faces of a coset complex — Lemma 1.2. In the coset complex of a group with subgroups K_i, the faces whose vertices have the types in a set A correspond to the left cosets of the intersection of the K_i with i in A.
• Purity, the action and the number of chambers — Proposition 1.3. A coset complex is pure; the group acts transitively on the faces of each type with stabilizers conjugate to the intersections; the chambers through a face of type A correspond to the cosets of the total intersection inside a coset of the intersection for A.
• Connectedness and generation — Proposition 2.1. A coset complex is connected if and only if the group is generated by the subgroups.
• Quotients of coset complexes — Proposition 3.1. If a normal subgroup is divided out and the image of every intersection of the subgroups is the intersection of the images, the faces of the coset complex of the quotient group are the orbits of the normal subgroup on the faces of the coset complex; the action is free on faces if the normal subgroup meets every subgroup trivially.
• The groups generated by elementary matrices on a cycle — Proposition 1.2. For a proper set J of edges of the cycle, the group generated by the corresponding elementary matrices consists exactly of the matrices with ones on the diagonal whose entry at (a, b) is in k, in kt, or zero, according to whether the path from a to b lies in J without or with the edge 0, or does not lie in J.
• Orders, intersections and reduction of the local groups — Corollary 1.3. The groups U_J have the orders q^p(J), satisfy U_J n U_K = U_(J n K), and reduction modulo t^m with m >= 2 is injective on them and preserves these intersections.
• Generation of the group U — Theorem 2.1. For N >= 3 and a prime q, the group of the N x N matrices over F_q[t] of determinant 1 that are upper unitriangular at t = 0 is generated by the elementary matrices e_(i,i+1)(1), 1 <= i < N, and e_(N,1)(t).
• The complexes T_m and X_m — Theorem 3.2. The coset complex T_m is finite, pure and connected with chambers corresponding to the group elements; a vertex lies in q^(N(N-1)/2) chambers and a face of codimension one in q chambers; the d-skeleton X_m has at most q^(N(N-1)/2) binomial(N-1, d) faces of dimension d at every vertex and at least N q^m / q^(N(N-1)/2) vertices.
• Coboundary expansion from two statements — Theorem 4.1. If the skeletons of dimension 2d - 1 of the complexes T_m satisfy an expansion inequality relative to the cocycles with a constant eta independent of m, and the cohomology of T_m with coefficients in the field with two elements vanishes in the degrees 1 to d - 1, then every X_m is a coboundary expander with the constant eta / q^(N(N-1)/2).

Reading order

Three lessons with complete proofs. They construct the complexes of the theorem and prove their elementary properties, and they reduce the theorem to two statements about these complexes that are named as hypotheses: one is a theorem of Oppenheim and Valentiner-Branth, the other is proved in OpenAI's text with buildings and the homology of groups.