Bounded-degree coboundary expanders in every dimension.
A study guide for a patient high-school reader, with the full proof.

How to read this file. This is a study guide to one research paper: the
construction, for every integer d >= 3, of finite d-dimensional simplicial
complexes of growing size, with a uniform bound on how many d-dimensional
faces can contain a single vertex, and with uniform F_2 coboundary
expansion in every degree below d. The goal is to give the reader the
full proof, not only the statement. The writing is deliberately
self-contained: each block begins with "Exercise:" followed by a short
question, then a self-contained answer. A "Proof:" inside an answer can
be skipped on a first read and returned to later. Named results carry a
short nickname in parentheses, e.g. "Lemma (Local Groups)"; find the
full statement later in this file by searching for the same nickname.

Map of the whole argument. The proof has five steps.
  Step 1. Build finite n-dimensional complexes T_m for n = 2d, whose
  local structure is "the same everywhere" but whose size grows with m.
  Step 2. Identify an infinite version L of these complexes with the
  subcomplex "opposite a chamber" in an affine building, and prove L
  is (n-2)-acyclic in reduced homology.
  Step 3. Use stabilization of group homology to prove that each finite
  T_m has vanishing reduced cohomology in degrees 1 through d - 1.
  Step 4. Apply a known quantitative theorem (from Kac-Moody-Steinberg
  groups) to get a lower bound on the coboundary of any cochain on the
  (n-1)-skeleton of T_m, uniformly in m.
  Step 5. Compare the intrinsic weight norms on the (n-1)-skeleton and
  on the d-skeleton, and use Step 3 to replace cocycles by coboundaries.
  That completes the proof.

Notation. Symbols you will meet early.
  d          ambient dimension; integer >= 3
  n = 2d     dimension of the intermediate complex T_m
  N = n + 1  = 2d + 1, number of vertices in each top face of T_m
  S          = {0, 1, ..., N-1}
  𝔽_q        finite field with q elements, q a large odd prime
  𝔽_2        finite field with two elements; our coefficient field
  k          = 𝔽_q, a shortcut
  R          = k[t], polynomials in t over k
  SL_N(R)    N by N matrices over R of determinant 1
  U          subgroup of SL_N(R): matrices that reduce at t = 0 to
             upper unitriangular matrices (upper triangular with 1 on
             the diagonal)
  Γ_a(m)     = kernel of SL_a(R) -> SL_a(R/(t^m)), the "congruence
             subgroup" of matrices congruent to the identity mod t^m
  Γ_N(m)     the case a = N
  G_m        = U / Γ_N(m), the finite quotient
  H_i        a "local group" for each i in S, defined in Step 1
  x_i        simple root generators of U, defined in Step 1
  U_J        for J subset of S, the subgroup of U generated by x_j
             with j in J
  T_m        a finite n-dimensional complex called the "coset complex"
  X_m        the d-skeleton of T_m; our final finite complex
  Y_m        the (n-1)-skeleton of T_m
  w_i, ||f||_i, Z^i, B^i, δ_i, dist    -- defined below

Foundations.

Exercise: What is an expander graph?
An expander graph G is a graph whose edge-to-vertex structure is
simultaneously sparse and well-connected. Precisely, for every subset
S of its vertices of size at most half the total, the boundary of S
(the set of edges leaving S) is a fixed fraction of |S|. Equivalently,
the second eigenvalue of the graph Laplacian is bounded below by a
positive constant independent of the graph size. The point is that
information cannot hide in any small region: any small set has a large
number of edges leaving it. Expander graphs are everywhere in computer
science: derandomization, error-correcting codes, network design.
Lubotzky 1994, Discrete Groups, Expanding Graphs and Invariant Measures:
https://doi.org/10.1007/978-3-0346-0398-0

Exercise: What is a simplicial complex?
A simplicial complex Y is a set of finite subsets of a vertex set Y(0),
called faces, with the rule that any subset of a face is also a face.
The dimension of a face σ is |σ| - 1. The dimension of Y is the largest
face dimension. Y is pure of dimension s if every face is contained
in an s-face. The s-faces are called chambers or top faces. For
0 <= i <= s, Y(i) is the set of i-faces. Vertices = Y(0), edges = Y(1).
Two chambers are adjacent if they share an (s-1)-face. The
"i-skeleton" of Y, written skel_i(Y), is the subcomplex consisting of
all faces of dimension at most i.

Exercise: Chains, cochains, boundary, coboundary.
Fix a coefficient field F, here F = 𝔽_2 (the field with two elements,
which is important: 1 + 1 = 0). The chain group C_i(Y; F) is the
F-vector space with basis Y(i). The boundary map
∂_i: C_i(Y; F) -> C_{i-1}(Y; F) sends an i-face σ = {v_0, ..., v_i} to
the sum, over the (i-1)-faces contained in σ, of those faces. Over
𝔽_2 there is only one sign, so each face appears with coefficient 1.
A key identity: ∂_{i-1} ∂_i = 0. So the image of ∂_i sits inside the
kernel of ∂_{i-1}; cycles = kernel(∂_i); boundaries = image(∂_{i+1});
homology H_i(Y; F) = cycles / boundaries. Reducing modulo boundaries
counts the "essential" cycles.

Dually, a cochain f is a function Y(i) -> F, equivalently an F-linear
functional on C_i(Y; F). The coboundary operator
δ_i: C^i(Y; F) -> C^{i+1}(Y; F)
is the transpose of ∂_{i+1}, so for τ in Y(i+1),
(δ_i f)(τ) = ∑_{σ ⊆ τ, |σ| = i+1} f(σ) in F.
Equivalently, for the characteristic function f = 1_A of a subset A of
i-faces, (δ_i f)(τ) is 1 if τ contains an odd number of faces of A,
else 0. Cocycles are Z^i(Y) = ker δ_i; coboundaries are B^i(Y) = im δ_{i-1};
cohomology is H^i(Y; F) = Z^i / B^i.

Exercise: Why are Z^i and B^i distinguished? (Coboundary vs cocycle.)
By definition B^i ⊆ Z^i, so H^i measures the gap. In graph terms, a
cocycle with values in 𝔽_2 is a colouring of vertices such that every
edge has colour sum zero; a coboundary is a "potential" colouring of
the form g(v) - g(u) along edges. In general, "cocycle" is a local
condition (the sum around every face vanishes), while "coboundary" is
a global condition (some lower-degree cochain produces it). The gap
H^i(Y; F) measures global topology. Any expansion statement against Z^i
alone allows nonzero cohomology classes to be "invisible" (they have
zero coboundary but positive distance from 0). A coboundary-expansion
statement must reduce distance against B^i, not against Z^i; that is
exactly why the theorem we prove must supply vanishing cohomology.

Exercise: What is a weighted norm and distance on cochains?
On a finite pure s-dimensional Y, define the weight of an i-face σ:
w_i^Y(σ) = #{s-faces τ ∋ σ} / (binom(s+1, i+1) · |Y(s)|).
The interpretation: pick an s-face uniformly at random, then pick one
of its (i+1)-vertex subsets uniformly; w_i^Y(σ) is the probability
that σ comes out. For a cochain f in C^i(Y; F),
||f||_{Y,i} = ∑_{σ : f(σ) ≠ 0} w_i^Y(σ),
which is the probability that f is nonzero at the sampled i-face. This
is exactly the right way to turn an F_2-valued function into a
"measure of support": it counts f modulo scaling by a uniform factor.

For any subspace A ⊆ C^i(Y; F), define
dist_{Y,i}(f, A) = min_{a ∈ A} ||f - a||_{Y,i}.
Taking A = B^i(Y) gives the natural distance to coboundaries that
appears in the main theorem.

Exercise: State the main theorem.
For every integer d >= 3, there exist constants D < ∞ and ε > 0,
together with a sequence of finite connected pure d-dimensional
simplicial complexes X_m with |X_m(0)| → ∞, such that:
  (a) every vertex belongs to at most D faces of dimension d;
  (b) for every m and every integer i with 0 <= i < d, and every
      cochain f in C^i(X_m; 𝔽_2),
        ||δ_i f||_{X_m, i+1}  >=  ε · dist_{X_m, i}(f, B^i(X_m)).
The constants D and ε do not depend on m, i, or f.

Condition (a) is "bounded local structure": any vertex sits in a
bounded number of top faces, so the complex is sparse. Condition (b)
is "uniform expansion": every cochain either has small support (is
close to a coboundary) or has a large coboundary. This is what is
called 𝔽_2 coboundary expansion (in every degree below the top).

Exercise: Why is this hard to prove?
Three separate difficulties.
1. It is easy to make a big complex with bounded vertex degree if you
let the lower skeleton be complete: random Steiner systems, Latin
squares, and similar constructions give expansion but the number of
top faces through a vertex grows with the size. Getting both "grows
with m" and "bounded vertex degree" at the same time needs a specific
group construction.
2. Vanishing of cohomology, H^i(X_m; 𝔽_2) = 0 for 1 <= i <= d - 1, has
to hold on the SAME complexes that have expansion. High-dimensional
"cosystolic expansion" gives only a lower bound on distance-to-cocycles,
not distance-to-coboundaries. A nonzero cohomology class is a cocycle
with zero coboundary and positive distance from 0, so it survives a
cosystolic bound but breaks coboundary expansion. A separate argument
is needed to kill cohomology, and it has to work in every degree.
3. The constants D and ε must be the same at every scale m. This needs
a uniform algebraic source of expansion, not a "the complex is big
enough" statement. The source here is a KMS (Kac-Moody-Steinberg)
cosystolic theorem for group coset complexes.

Exercise: What is a coset complex?
Let G be a group, and (K_0, ..., K_{N-1}) a finite list of subgroups
of G. The coset complex CC(G; (K_i)_{i in S}) has vertices which are
"tagged left cosets" g K_i with g in G and i in S = {0, ..., N-1}.
A collection of vertices is a face iff the corresponding cosets have
a common intersection: {g_0 K_{i_0}, ..., g_r K_{i_r}} is a face iff
some h in G lies in ∩_j g_j K_{i_j}. The chambers (top faces) are the
sets {g K_i : i in S}, one per element of G modulo the common
stabilizer N = ∩_i K_i. The complex is pure of dimension |S| - 1 = N - 1.

Exercise: Why coset complexes give bounded vertex degree.
For a fixed face F = {g K_i : i in A} of type A ⊆ S, the chambers
through F are the chambers of the form g K ∩ ... indexed by a coset of
the intersection of the K_i for i ∉ A. Precisely: the number of
chambers through F equals |U_{S \ A}| where U_J is the intersection
of the K_i for i in J. So if the pairwise intersections U_J are
finite groups of bounded size, the number of top faces through any
face is bounded. In our construction, N = ∩ K_i is trivial (so
chambers are in bijection with G itself) and U_J is a fixed finite
group, giving uniform bounds independent of m.

Part 1. The congruence complexes.

Exercise: Setup. Define N, S, x_i, H_i.
Let d >= 3, and set
  n = 2d,   N = n + 1 = 2d + 1,   S = {0, 1, ..., N - 1}.
Let q be a sufficiently large odd prime, to be chosen once and for all
near the end (all largeness requirements will be collected there).
Set k = 𝔽_q, R = k[t],
  Γ_a(m) = kernel of SL_a(R) -> SL_a(R / t^m).
Define U ⊆ SL_N(R) as the inverse image, under "evaluation at t = 0",
of the group of upper unitriangular N × N matrices over k. So U is
the set of matrices in SL_N(k[t]) whose mod-t reduction is upper
triangular with 1's on the diagonal.

For each i in S define a "simple generator"
  x_i = e_{i, i+1}(1)   for 1 <= i < N,
  x_0 = e_{N, 1}(t),
where e_{a,b}(f) is the N × N matrix equal to the identity except at
the (a, b) entry, which equals f. (Row and column indexing 1 to N.)
Then define H_i = <x_j : j ∈ S, j ≠ i>, the subgroup of U generated
by all the simple generators other than x_i.

Exercise: Why does x_0 carry a t and the others do not?
Look at S as the vertices of a directed cycle: edge i from i to i+1
(mod N). Edge 0 is the wrap-around edge from N to 1. The generator
x_0 lies on this wrap-around edge and needs a factor of t to stay
inside SL_N(k[t]) (otherwise e_{N,1}(1) would have determinant 1 but
would not reduce to an upper triangular matrix at t = 0). The other
generators x_1, ..., x_{N-1} sit on the "non-wrap" edges of the cycle
and are already upper triangular at t = 0.

This cycle structure is the whole point. The simple generators x_i
satisfy the relations of an affine root system of type Ã_n (the cyclic
Dynkin diagram with N nodes). That is the KMS group structure. If we
worked with the path 1 -> 2 -> ... -> N instead of the cycle (i.e.,
with U being just the upper unitriangular matrices over k[t]), we
would not be in the affine Kac-Moody setting, and the cosystolic
theorem of Step 4 would not apply.

Exercise: What is V_ab(J)?
View S as the boundary cycle 1 -> 2 -> ... -> N -> 1, so that edge
i is the edge from vertex i to vertex i+1 (mod N), with the special
case that edge 0 is the wrap-around edge N -> 1. For J ⊊ S, we say a
directed path "lies in J" if every edge of the path is in J. Between
any two distinct indices a, b there is exactly one simple directed path
(no repeated vertices) in the cycle. Define V_ab(J) as the following
subspace of R = k[t]:
  - k,  if the directed path from a to b lies in J and avoids edge 0;
  - kt, if the directed path from a to b lies in J and uses edge 0;
  - 0,  otherwise (some edge of the path is not in J).
Here k and kt are the degree-0 and degree-1 pieces of R.

The point of V_ab(J) is that it encodes "which paths through the group
are turned on by J". The trivial generator x_i = e_{i,i+1}(1) turns
on the entry (i, i+1) at degree 0; the wrap generator x_0 = e_{N,1}(t)
turns on the entry (N, 1) at degree 1. Composing these, via commutators,
produces every path entry, but only along edges that lie in J.

Exercise: Lemma (Local Groups). State and prove.
For every proper J ⊊ S, put U_J = <x_j : j in J>. Then
  U_J = {g in SL_N(R) : g_{a a} = 1 for all a; g_{a b} in V_{ab}(J) for
         all a ≠ b}.
These U_J are finite groups, and the following intersection identities
hold:
  U_J ∩ U_K = U_{J ∩ K} for all J, K ⊊ S,
  ∩_{i in A} H_i = U_{S \ A} for any nonempty A ⊆ S.
For m >= 2, the reduction R -> R / t^m is injective on every U_J and
preserves these intersections. In particular,
  |H_i| = M := q^{binom(N, 2)} for every i in S, and |U_{i}| = q.

Sketch of proof.
(a) Cut the cycle. Choose j_0 ∉ J. Deleting the edge j_0 turns the
cycle into a linear order (j_0+1, j_0+2, ..., j_0+N-1) on the vertices
of S. In this linear order, the group {g : g_aa = 1, g_ab in V_ab(J)}
is a product of unitriangular groups, one for each path component of J,
the product taken in the linear order. It is a standard fact that
unitriangular groups over a commutative ring are groups, so the
displayed set is a group. This proves the "⊇" direction. Then note
that if any coefficient of g involves a t^k with k >= 2, the path
descriptions V_ab(J) = 0 or k or kt rule it out, so the group is over
k and kt = k[t]/(t^2). That is finite.

(b) "⊆" direction. Because q is prime, the unitriangular generator x_j
raised to various powers e_{j,j+1}(a) as a ranges over k produces every
coefficient in the simple root subgroup. Then iterated commutators
  [e_{a,b}(f), e_{b,c}(g)] = e_{a,c}(f g)  for a ≠ b ≠ c
allow composing paths: at each step you append an edge of the cycle.
Taking all such compositions and using unitriangular elimination in
the cutting order of (a) produces every matrix in the display. This
proves U_J equals the displayed set.

(c) Intersections. The entrywise conditions describing U_J make it
clear that U_J ∩ U_K = U_{J ∩ K}: a matrix is in U_J ∩ U_K iff its
pattern of nonzero entries conforms to both J and K, which is exactly
J ∩ K. For H_i = <x_j : j ≠ i>, the intersection over A ⊆ S of H_i
is the subgroup generated by the x_j with j ∉ A, i.e., U_{S \ A}.

(d) Reduction mod t^m (m >= 2). Every element of every U_J has entry
in k or kt (no t^2 terms), so reducing mod t^m for m >= 2 neither
identifies two distinct elements nor changes which entries are
allowed. This gives injectivity and preserves the intersections.

(e) |H_i| = q^{binom(N, 2)}. Each U_J for J = S \ {i} and count: the
number of (a, b) with a ≠ b such that V_ab(J) is either k or kt. In
the linear order obtained by deleting edge i, the pairs (a, b) with
V_ab(J) ≠ 0 are exactly the pairs with a to the left of b in that
linear order, i.e., exactly binom(N, 2) of them. Each such pair
contributes a factor of q. So |U_J| = q^{binom(N, 2)} = M. Similarly
U_{i} = <x_i> has order q (its generator has order q; x_0 has order q
too since q is odd and t has infinite order, but the mod-t reduction
still gives q distinct images in SL_N(k)).

Exercise: Why n = 2d, exactly?
Two constraining inequalities force it.
(i) Expansion theorem (Step 4) is proved for n-dimensional, n-classical
KMS complexes with n >= 3, and gives a bound on the (n-1)-skeleton.
For coboundary expansion in the theorem at dimension d, we need to
land on X_m = skel_d T_m, and this must be a lower-dimensional skeleton
of T_m; we need d < n-1, i.e., d <= n-2, i.e., n >= d + 2.
(ii) Cohomology vanishing (Step 3) uses two stabilization steps and
needs N - 2 >= 2(d - 1) + 1. That is N - 2 >= 2d - 1, i.e., N >= 2d+1.
With N = n + 1, this is n + 1 >= 2d + 1, i.e., n >= 2d.
So the two conditions are n >= d + 2 and n >= 2d. For d >= 2 the
second is stronger, so n = 2d is the smallest choice satisfying both.
We use exactly n = 2d, N = 2d+1.

Exercise: Define T_m and X_m, the finite congruence complexes.
For m >= 2, set
  G_m     = U / Γ_N(m)  (finite, because Γ_N(m) is finite-index in U),
  T_m     = CC(G_m; (H̄_i)_{i in S}),
  X_m     = skel_d T_m,
where H̄_i is the image of H_i in G_m. The complex T_m has dimension
n = N-1, is pure, and X_m has dimension d. Since H_i injects into
G_m (Lemma Local Groups says reduction mod t^m is injective on U_J for
m >= 2), the local groups of T_m are the same as those of U: |H̄_i| = M
for each i.
The infinite version L := CC(U; (H_i)_{i in S}) will be our Step 2
object. Its simplex stabilizers are conjugates of the U_J, which are
finite groups of odd order (q is odd), so it satisfies the hypotheses
of the "Projective Chains" lemma below.

Exercise: Why is every simplex stabilizer of L of odd order?
Every nonempty face of L of type A ⊆ S has stabilizer a conjugate of
∩_{i in A} H_i = U_{S \ A}, hence (by Lemma Local Groups) a finite
unitriangular group over k = 𝔽_q for some q that will be a sufficiently
large odd prime, so every group element has order a power of q. Since
q is odd, that is odd. (We do not need more than that the order is
odd — finite and odd is the whole requirement.)

Part 2. The infinite complex L as an opposition subcomplex.

Exercise: What is a building? (just enough for what we need)
A building of type (W, S) is a simplicial complex Δ whose maximal
simplices (called chambers) are all of the same size |S|, whose faces
are coloured by S subject to the following rules:
  - Every chamber has exactly one face of each type s in S.
  - Any two chambers are joined by a gallery, i.e., a sequence
    c_0, c_1, ..., c_k of chambers with c_i and c_{i-1} sharing an
    (s_i-1)-face (an s_i-panel), for a word s_1 s_2 ... s_k in W.
  - This gallery is minimal for exactly one word in W (the "Weyl
    distance" between the two chambers). Panels around a chamber obey
    the Coxeter relations of W.
Here (W, S) is a Coxeter group: W = <S | s^2 = 1, (s s')^{m_{s s'}} = 1>.
If some product of generators m_{s s'} is infinite, W is "infinite type".
The most important infinite-type case is the affine Coxeter group of
type Ã_n: its generators S = {0, ..., N-1} with the cyclic Dynkin
diagram (path 1 - 2 - ... - N with an extra edge N - 1). Its dimension
is N-1 = n.
Abramenko-Brown 2008, Buildings: Theory and Applications:
https://doi.org/10.1007/978-0-387-78835-7

Exercise: The affine twin building of SL_N.
Let G = SL_N(k[t, t^{-1}]) with the diagonal torus T of constant
diagonal matrices. Define two parabolics
  B_+ = {g in G : g(t) in SL_N(k[t]) and g(0) is upper triangular},
  B_- = {g in G : g(t^{-1}) in SL_N(k[t^{-1}]) and g(0) is lower
        triangular},  analogously at "t = ∞".
Both are Borel subgroups of G, and B_+ = U ⋊ T (semidirect: T is
normal, and U is the unipotent radical, exactly the U from Step 1,
embedded in G). Also B_+ ∩ B_- = T.

Form the simplicial complex Δ_+ whose chambers are left B_+-cosets
in G; faces are parabolic cosets inside a chamber. By Tits's theorem,
Δ_+ is a building of type Ã_{N-1} ≅ Ã_n. Similarly Δ_- from B_-.
Together they form an "affine twin building of type Ã_n"
(Abramenko-Van Maldeghem 1-twinnings, 2001):
https://doi.org/10.1007/PL00004899
The "twin" structure is a codistance map
  δ*: {chambers of Δ_-} × {chambers of Δ_+} → W ∪ {∞}
with axioms: δ*(c, c') = δ*(c', c) (symmetric); δ*(c, c_+) = 1 for
fixed positive base chamber c_+ iff c is in the B_+-orbit of the
negative base chamber c_-; and if c' is a chamber adjacent to c along
an s-panel, then δ*(c', c_+) ∈ {δ*(c, c_+), δ*(c, c_+) s} with ws
attained. When ℓ(w s) < ℓ(w), every chamber c' in that panel other
than c has δ*(c', c_+) = w s.

Exercise: What is L' (the opposition subcomplex)?
Let c_+ be the positive base chamber (the B_+-coset at 1). Let c_- be
the unique negative base chamber with δ*(c_-, c_+) = 1. Define
  L' = (all chambers c of Δ_- with δ*(c, c_+) = 1) plus all their faces.
This is a subcomplex of Δ_-, because if a chamber has type-specific
vertices, its faces also do. Everything here is type-A: vertices of Δ_-
are parametrized by homothety classes of k[[t^{-1}]]-lattices L ⊂ k[[t^{-1}]]^N
of rank N. The base chamber c_- has vertices with exponent vectors
  type 0:    b = (0, 0, ..., 0),
  type i:    b = (0, ..., 0, 1, ..., 1)  (i zeros, N-i ones).

Exercise: Lemma (Opposite Stabilizers, aka L' ≅ L). State and prove.
U acts simply transitively on the chambers of L', and the type-i vertex
stabilizers of the base chamber are exactly H_i. Consequently,
  L := CC(U; (H_i)_{i in S}) ≅ L'.
Equivalently, there is an isomorphism of simplicial complexes sending
u H_i to the vertex of L' obtained by applying u to the type-i vertex
of the base chamber.

Proof.
(a) U-transitivity on codistance-1 chambers. By the twin-building
axiom, chambers of Δ_- of codistance 1 from c_+ form a single B_+-orbit,
namely B_+ · c_-. Since B_+ = U · T and T stabilizes c_-, U already
acts transitively on them. If u c_- = c_- for some u in U, then
u is in the stabilizer of c_-, which is B_- ∩ G^{-1} ... (specifically,
B_- ∩ (chambers-opposite-c_+ stabilizer)); intersecting with B_+ = U ⋊ T
gives T, and U ∩ T = {1}. So U acts simply transitively on the
codistance-1 chambers. ✓

(b) Vertex stabilizers of c_- in U. The elements of G acting on Δ_-
preserve a vertex b (a homothety class of lattices) exactly when
g_{ab} satisfies
  deg(g_{ab})  ≤  b_a - b_b   for all a, b
Here deg is the t-adic degree over k[t], and the bound comes from
requiring g to map the standard basis t^{b_a} e_a to something still in
the lattice (up to scalar). The bound is the same for the negative
building as for the positive: at type 0 (all b_a = 0), we get
deg(g_{ab}) ≤ 0, so g_{ab} ∈ k. Combined with U being upper
unitriangular at t = 0, this gives g_{aa} = 1 and g_{ab} ∈ k for a < b
and g_{ab} = 0 for a > b: exactly the constant upper unitriangular
matrices, i.e., H_0.

At type i ≠ 0, cut after coordinate i so that the vertex exponent is
  (b_1, ..., b_N) = (0, ..., 0, 1, ..., 1) with i zeros.
Then the bound deg(g_{ab}) ≤ b_a - b_b gives:
  a, b ≤ i:       deg ≤ 0 ⇒ g_{ab} ∈ k, together with U: upper
                  unitriangular constant block of size i × i
                  (equivalently (i+1) × (i+1) with 1's on diagonal).
  a ≤ i < b:      deg ≤ -1 with g_{ab} ∈ k[t] ⇒ g_{ab} = 0 (top-right
                  block vanishes).
  a > i, b ≤ i:   deg ≤ 1 with U (g_{ab}(0) = 0) ⇒ g_{ab} ∈ kt
                  (lower-left block, arbitrary entries in kt).
  a, b > i:       deg ≤ 0 ⇒ g_{ab} ∈ k, upper unitriangular constant
                  block of size (N-i) × (N-i) with 1's on diagonal.

The result matches the pattern described by Lemma (Local Groups) for
J = S \ {i}: two upper unitriangular constant blocks, zero top-right,
arbitrary kt lower-left. So the U-stabilizer of the type-i vertex of
c_- is exactly H_i. ✓

(c) Assembling the isomorphism. Send u H_i to the type-i vertex of the
chamber u c_-. By (b), this vertex depends only on u H_i (left u acts
on chambers, and the base-chamber identification is through H_i). By
(a), the map is a bijection on 0-simplices; faces of L correspond to
common representatives, and by (b) these are exactly L' faces. So
L ≅ L' as simplicial complexes.

Part 2 (continued). The codistance filtration of the negative building.

Exercise: Set up the filtration.
We filter the negative building Δ_- by adding chambers of Δ_- in increasing
order of the codistance length ℓ(δ*(c, c_+)). Step 1 is exactly L' = the
codistance-1 chambers and their faces. Step k adds all chambers whose
codistance has length exactly k, together with their faces. The whole
negative building is the increasing union.
We need the following fact: length function on W satisfies
  |ℓ(ws) - ℓ(w)| = 1 for all s in S.
(Standard fact in Coxeter groups.)

Exercise: Set up one codistance step.
Fix a codistance value w ≠ 1 in W (not the identity). Define the descent
set and its complement:
  D  = {s in S : ℓ(ws) < ℓ(w)},   J = S \ D.
Since w ≠ 1, D is nonempty (every nontrivial element of W has a right
descent), so J ⊊ S. As the affine Dynkin diagram is a cycle, deleting
any node turns it into a finite path, so the parabolic W_J is a finite
spherical Coxeter group (a product of A_r components). Let w_J be the
longest element of W_J.

The minimal coset representative: since w has no descents in J, we have
  ℓ(w v) = ℓ(w) + ℓ(v) for all v in W_J,
and w is the unique minimal representative of the coset w W_J. In
particular, ℓ(w · w_J) = ℓ(w) + ℓ(w_J) is the longest length in w W_J.

Exercise: Lemma (Block Structure). Describe one codistance step.
For each codistance call w (as above), the codistance-w chambers are
grouped into "blocks" indexed by J-residues. Each J-residue R is a
subcomplex of Δ_- fixed outside type-J positions: all faces of a fixed
D-type are common to every chamber of R, and the remaining types vary
inside a spherical building of type J. Within one block, the codistance-w
chambers are exactly the chambers of R opposite a single chamber
c_max ∈ R (of codistance w · w_J). So the codistance-w chambers of R,
together with their faces, form the simplicial join
  F * O
where F is the common D-face of the residue and O is the opposition
complex of c_max inside R (a spherical opposition complex of type J).

Proof. (a) Step up from any chamber of R. Fix a codistance-w chamber
c of R, and s in J. The s-panel through c has q+1 chambers (a spherical
residue of type A_1 has q+1 chambers when q = |𝔽_q|). By the codistance
panel axiom, the two possible codistances of these chambers are w and
ws, one being of shorter length. Since s ∈ J \ D, ℓ(ws) > ℓ(w), so s is
an ascent. Therefore c is the SHORTEST chamber of its s-panel, and the
other q chambers are at codistance ws.
Iterating along a reduced expression for w_J (length ℓ(w_J) steps, all
in J hence all ascents), from c we reach a chamber c_max of R at
codistance w · w_J. This chamber does not depend on the choice of
reduced expression by the standard "well-definedness of codistance
from c_+" property of twin buildings.

(b) Codistance-decreasing galleries from c_max. Any minimal gallery in
R from c_max to a chamber d has type-word equal to a reduced word for
the Weyl distance v ∈ W_J between c_max and d. Applying the descent
panel axiom at each step (all s in J are descents of w · w_J, since
ℓ(w · w_J · s) = ℓ(w) + ℓ(w_J) - 1 < ℓ(w · w_J)) gives:
  endpoint d has codistance w · w_J · v.
The endpoint d has codistance w iff w · w_J · v = w iff w_J · v = 1 in
W iff v = w_J (using w_J · w_J = 1, true since w_J is the longest of a
finite Coxeter group and thus its own inverse). So the codistance-w
chambers of R = the chambers at Weyl distance w_J from c_max = the
opposition complex of c_max in R. ∎

(c) Attaching loci. Within F * O, the faces that contain every vertex of
F are exactly the codistance-w chambers of R and their subfaces. Any
such face was not previously added: no face containing all of F could
have come from a chamber of a DIFFERENT residue, because faces outside
such a locus determine the D-face F. A face that omits some vertex of
F lies in the boundary ∂F of F, hence in a D-descent panel, hence in a
codistance-ws chamber for some s ∈ D (which has shorter length and was
added earlier). So the intersection of F * O with the previous subcomplex
is exactly ∂F * O.

Exercise: Lemma (Block Homology). Relative chains of one block.
For a block F * O as above, the relative chain complex
C_k(F * O, ∂F * O; 𝔽_2) is isomorphic to the augmented chain complex
C̃_{k - |D|}(O; 𝔽_2), shifted by |D|. So
  H_k(F * O, ∂F * O; 𝔽_2)  ≅  H̃_{k - |D|}(O; 𝔽_2).
Reason: in the relative chain complex, a face is zero if it lies in the
subcomplex ∂F * O. A face of F * O is F' * σ with F' ⊆ F and σ ⊇ ∅.
It is in ∂F * O iff F' ⊊ F. So only faces with F' = F survive. Among
these, deleting a vertex of F ≥ F * σ counts as zero in the relative
complex, and the remaining boundary ∂σ ∈ C̃_{|σ| - 2}(O) is exactly the
augmented boundary of O. So relative chains = shifted augmented chains
of O. ∎

Exercise: Input (Spherical Opposition Vanishing).
Let Δ be a spherical building of type A_r over 𝔽_q with q >= 2^{r-1}.
Let c be a chamber of Δ and O the complex of chambers opposite c,
together with their faces. Then
  H̃_j(O; 𝔽_2) = 0  for  j < r - 1.
This is one of the consequences of the "integral cone function" chain
contraction of Oppenheim-Valentiner-Branth, Corollary 6.10 in
[OVB] (Oppenheim-Valentiner-Branth, arXiv:2504.05823v2). The chain
contraction identity is Z-linear and remains valid after reducing mod 2,
provided q = |𝔽_q| is at least 2^{r-1}.
For a product Δ = Δ_1 × ... × Δ_k of spherical buildings of type
A_{r_1}, ..., A_{r_k}, the opposition complex is the join
O_1 * ... * O_k of the factor opposition complexes. The augmented chain
formula for joins gives vanishing below the top dimension:
  H̃_j(O; 𝔽_2) = 0  for  j < (r_1 + ... + r_k) - 1 = dim O.
Here dim O = dim Δ = sum r_i - k. In our setup each J-residue of Δ_- is
a product of type-A buildings of total rank |J| = |S| - |D|, so
dim O = |J| - 1. We henceforth require q >= 2^{n-1}; this covers every
proper residue of Δ_- (since |J| <= n, so |J| - 1 <= 2^{n-2} <= 2^{n-1}).

Exercise: Proposition (L is (n-2)-acyclic). Prove H̃_j(L; 𝔽_2) = 0 for
0 <= j < n - 1.

Proof.
(1) Apply the Block Homology lemma in each codistance step. The relative
homology of one block F * O vanishes below degree |D| + dim O = |D| +
|J| - 1 = |S| - 1 = n. Summing over all blocks in one codistance step
(a direct sum), the relative homology of the entire step vanishes for
k < n.

(2) Consecutive steps. Consider the pair (T_{< w}, T_{<= w}) where
T_{< w} = all chambers of codistance < w (with their faces) and
T_{<= w} = T_{< w} ∪ (codistance-w chambers). By the attaching-locus
computation, the pair (T_{<= w}, T_{< w}) is filtered by the blocks,
and its relative homology in degree k vanishes for k < n.
The long exact sequence in homology of the pair (T_{<= w}, T_{< w}) has
  ... -> H̃_j(T_{< w}) -> H̃_j(T_{<= w}) -> H̃_j(pair) -> H̃_{j-1}(T_{< w}) -> ...
For j <= n - 2, both H̃_j(pair) and H̃_{j+1}(pair) vanish, so the map
H̃_j(T_{< w}) -> H̃_j(T_{<= w}) is injective (from j+1 vanishing) and
surjective (from j vanishing), hence an isomorphism.

(3) Pass to the limit. The full negative building Δ_- is the increasing
union of T_{<= w} over codistance values w, and simplicial homology over
a field commutes with such increasing unions:
  H̃_j(Δ_-; 𝔽_2) = colim_w H̃_j(T_{<= w}; 𝔽_2).
By (2) applied successively, all transition maps
H̃_j(T_{< w}) -> H̃_j(T_{<= w})  (for j <= n - 2) are isomorphisms.
So H̃_j(Δ_-; 𝔽_2) = H̃_j(L; 𝔽_2) for j <= n - 2.

(4) Contractibility of Δ_-. The affine building Δ_- is a Euclidean
building of type Ã_n, hence a CAT(0) space, hence contractible.
Specifically, Bridson-Haefliger, Metric Spaces of Non-Positive Curvature
(1999), Theorem II.10A.4 and Remark II.10A.6. So H̃_j(Δ_-; 𝔽_2) = 0 for
all j, and therefore H̃_j(L; 𝔽_2) = 0 for 0 <= j < n - 1. ∎

Exercise: Corollary (Generation). The simple generators x_i generate U.
Proof. By the acyclicity proposition, L is (n-2)-acyclic; in particular
H̃_0(L; 𝔽_2) = 0, so L is connected. In a connected pure complex, any
two chambers can be joined by a sequence of chambers in which successive
chambers share a vertex. Label chambers of L by u in U (via the U-action,
which is simply transitive on chambers by Lemma Opposite Stabilizers).
If chambers labelled u and v share a vertex of type i, then u H_i = v H_i,
so u^{-1} v ∈ H_i. Walking a chamber-path from the base chamber 1 to a
chamber u gives u = h_{i_1} h_{i_2} ... h_{i_k} with each h_{i_j} ∈ H_{i_j}.
So U = <H_i : i in S>. Each H_i is generated by {x_j : j ≠ i} by
definition. ∎

Part 3. Homology and the finite quotients.

Exercise: What is group homology?
For a group K and an abelian group A with trivial K-action, H_j(K; A)
is the j-th left derived functor of coinvariants M ↦ M_K = M ⊗_{Z[K]} Z.
Concretely: take a projective resolution P_* → Z of the trivial module
Z over the group ring Z[K], tensor over Z[K] with A (giving A-valued
chains), and take homology. Equivalently, take a contractible K-space
with free K-action and quotient.
The reason this appears here: the complex L has a U-action with
nice stabilizers, so L "looks like" a contractible space modulo bounded
stabilizer corrections. That is exactly what the Lemma below formalizes.

Exercise: Lemma (Projective Chains). State and prove.
Let h ≥ 0. Let K act on a nonempty simplicial complex Z without
inversions. Suppose every nonempty simplex stabilizer is finite of odd
order. If H̃_j(Z; 𝔽_2) = 0 for 0 ≤ j ≤ h (i.e., Z is h-acyclic in
reduced homology), then
  H_j(K; 𝔽_2)  ≅  H_j(C_*(Z; 𝔽_2)_K)       for 0 ≤ j ≤ h,
where C_*(Z; 𝔽_2)_K means the chain complex of K-orbits of simplices
with boundary induced by deleting vertices, and H_j on the right is
ordinary homology.

Proof.
For each r ≥ 0, the permutation module C_r(Z; 𝔽_2) is a direct sum
of modules 𝔽_2[K/H] as H ranges over simplex stabilizers. Since |H| is
odd, averaging over H gives a splitting of the augmentation map
𝔽_2[H] → 𝔽_2. So 𝔽_2 is a projective 𝔽_2[H]-module. By induction
𝔽_2[K/H] = 𝔽_2 ⊗_{H} 𝔽_2[K] is a projective 𝔽_2[K]-module. Thus every
chain module C_r(Z; 𝔽_2) is projective as a 𝔽_2[K]-module.
The augmented chain complex C_*(Z; 𝔽_2) → 𝔽_2 is exact through degree
h by hypothesis. Truncate at degree h+1: the modules C_0, ..., C_{h+1}
together with differentials ∂_1, ..., ∂_{h+1} form a partial resolution.
Complete this to a full projective resolution by choosing (using the
projective cover property) a projective module P_{h+2} mapping onto
ker(∂_{h+1}) and repeating. The result P_* → 𝔽_2 agrees with C_*(Z; 𝔽_2)
in degrees 0, ..., h+1 with the SAME differentials.
Then:
  H_j(K; 𝔽_2) = H_j(P_* ⊗_K 𝔽_2) = H_j(C_j(Z; 𝔽_2) ⊗_K 𝔽_2)
              = H_j(C_j(Z; 𝔽_2)_K)
for 0 ≤ j ≤ h, since in this range the chain modules and differentials
of P_* and C_*(Z; 𝔽_2) agree, and coinvariants is a functor. ∎

Exercise: Proposition (Homology of U and of T_m). State the first half.
For 0 < j < n-1 and each m ≥ 2:
  H_j(U; 𝔽_2) = 0,     H_j(T_m; 𝔽_2)  ≅  H_j(Γ_N(m); 𝔽_2).
Also T_m is connected.

Proof.
Part I: H_j(U; 𝔽_2) = 0 for 0 < j < n-2+1 = n-1.
Apply the Projective Chains Lemma with K = U, Z = L, h = n - 2.
Hypotheses:
- Every nonempty face stabilizer in L is a conjugate of U_{S \ A} for
  some type A, hence (Local Groups lemma) a finite unitriangular group
  over k = 𝔽_q. Since q is odd, its order is odd. ✓
- L is (n-2)-acyclic by the Acyclicity Proposition. ✓
Conclusion: H_j(U; 𝔽_2) ≅ H_j(C_*(L; 𝔽_2)_U) for 0 ≤ j ≤ n-2.
The right side is the chain complex of the orbit complex U \ L, but
with the orbit boundary induced by deleting vertices. The orbit complex
U \ L has exactly one face of each type: U acts transitively on
type-A faces for any A ⊆ S (because for any type-A face, its stabilizer
is a conjugate of U_{S \ A}, and all type-A faces are such conjugates).
So U \ L is a single n-simplex, with one face of each type A ⊆ S.
The orbit chain complex equals the ordinary chain complex of this
n-simplex, with coefficient 1 on each boundary map (no stabilizer
corrections needed: we are in 𝔽_2). Hence H_j(C_*(L; 𝔽_2)_U) = 0 for
0 < j ≤ n-2. That proves the first assertion.

Part II: H_j(T_m; 𝔽_2) ≅ H_j(Γ_N(m); 𝔽_2) for 0 < j < n-1.
First, show L/Γ_N(m) ≅ T_m. The reduction map π: L → T_m sends g H_i
to π̄(g) H̄_i where π̄: U → G_m = U/Γ_N(m) is the quotient map, and H̄_i
denotes the image of H_i in G_m. This is well-defined: if g' H_i = g H_i
with g' = g h for h ∈ H_i, then π̄(g') H̄_i = π̄(g) H̄_i. It is bijective
on faces:
- Injective on faces by Local Groups: reduction mod t^m is injective
  on every U_J for m ≥ 2, so g H_{i_1} ∩ ... ∩ g H_{i_A} = (simplex)
  is preserved.
- Surjective on faces because π̄: U → G_m is surjective by the
  Generation Corollary (x_i generate U, so their images generate G_m).
Fiber over a face = Γ_N(m)-orbit. So L / Γ_N(m) ≅ T_m as simplicial
complexes.
Second, Γ_N(m) acts FREELY on L. Any nonempty face F of L has stabilizer
a conjugate of U_{S \ A}, so Γ_N(m) ∩ (conjugate of U_{S \ A}) = trivial
(since Γ_N(m) is normal in U and kills the image of U_J under reduction
mod t^m). So every face-orbit is a face.
Third, apply Projective Chains Lemma with K = Γ_N(m), Z = L, h = n - 2.
The complex L is (n-2)-acyclic, the stabilizers are trivial (in particular
finite of odd order — vacuous), so
  H_j(Γ_N(m); 𝔽_2) ≅ H_j(C_*(L; 𝔽_2)_{Γ_N(m)})
                  = H_j(C_*(Γ_N(m)\L; 𝔽_2))
                  = H_j(T_m; 𝔽_2)
for 0 ≤ j ≤ n-2. In particular for 0 < j < n-1.
Connectedness of T_m: L is connected (H_0(L; 𝔽_2) = 0 from Proposition
L-acyclic), hence its quotient Γ_N(m)\L ≅ T_m is connected. ∎

Part 3 (continued). From pro-stability to levelwise surjection.

The next result shows that obstruction-group homology at dimension j gets
simpler as the matrix size a grows. The input is a deep theorem about
inverse systems (Iwasa), and the output is a concrete levelwise statement
about the finite quotients Γ_a(m).

Exercise: What is the stable range of a nonunital ring?
For a commutative ring B (with or without unit), the stable range sr(B)
is the smallest s such that every "unimodular row" of length ≥ s+1 can
be shortened by an elementary operation to another unimodular row. This
is classical stability theory (Bass, Vaserstein, Suslin). Polynomial
rings over fields have small stable range, which is why we can use them.

The exact definitions we need are due to Vaserstein 1969 (Section 2,
pp. 384-385 in Math. USSR-Sbornik 8):
- For a commutative nonunital ring A, its INTEGER UNITIZATION is
  Z ⋉ A := Z ⊕ A, with multiplication
    (u, a)(v, b) = (uv, ub + va + ab),  identity 1 = (1, 0).
  The projection Z ⋉ A → Z has kernel A. Define
    GL_a(A) := ker(GL_a(Z ⋉ A) → GL_a(Z)).
- A row b ∈ (Z ⋉ A)^ℓ is A-unimodular if b ≡ e_1 mod A and there is
  c ∈ (Z ⋉ A)^ℓ with c ≡ e_1 mod A and b · c = 1.
- sr(A) ≤ s means: for every A-unimodular row b of length ℓ ≥ s+1,
  there exist v_1, ..., v_{ℓ-1} ∈ A such that
    (b_1 + v_1 b_ℓ, ..., b_{ℓ-1} + v_{ℓ-1} b_ℓ)
  is A-unimodular.

Exercise: What is a "pro-isomorphism" of inverse systems?
An inverse system of abelian groups is a sequence
  A_1 ← A_2 ← A_3 ← ...
with transition maps t_{m' m}: A_{m'} → A_m for m' ≥ m (compatible).
It is PRO-ZERO if, for every m ≥ 1, some sufficiently late transition
t_{m' m}: A_{m'} → A_m is the zero map.
A level map of inverse systems f: (A_m) → (B_m) is a
PRO-ISOMORPHISM iff its kernel and cokernel systems are both pro-zero.
The point: pro-isomorphism means "eventually looks like an isomorphism
inside the system", but NOT that each level map A_m → B_m is an
isomorphism at any FIXED m.

Definition (Tor-unital): an inverse system (A_m) of commutative
nonunital rings is Tor-unital if, for every i ≥ 1, the inverse system
(Tor_i^{Z ⋉ A_m}(Z, Z))_m is pro-zero.

Exercise: Theorem (Morrow; Iwasa). State the two deep inputs.
Morrow (J. reine angew. Math. 736 (2018), 95-139, Theorem 0.3 in
arXiv:1404.4179v2). If B is a commutative Noetherian ring and I ⊆ B
is an ideal, then the powers (I^m)_{m ≥ 1} form a Tor-unital inverse
system. (Noetherian = every ideal is finitely generated. The polynomial
ring k[t] is Noetherian, so the ideal (t^m) fits.)
In our case, with B = k[t] and I = (t), the system A_m := t^m k[t] is
Tor-unital.

Iwasa (Homology, Homotopy and Applications 22 (2020), no. 1, 343-374,
Theorem 5.13 in the published version). Let (A_m) be a Tor-unital inverse
system of commutative nonunital rings with
  r := max{2, sup_m sr(A_m)} < ∞.
Then for every j ≥ 0, the natural map
  (H_j(GL_a(A_m); Z))_m  →  (H_j(GL_{a+1}(A_m); Z))_m
is a pro-isomorphism for a ≥ 2j + r - 1.
The map is the one induced by inclusion of the first a coordinates
into the first a+1 coordinates.

Exercise: Lemma (Stabilization). State and prove.
Let h ≥ 0. If a ≥ 2h + 1, the inclusion on the first a coordinates
induces SURJECTIONS
  H_j(Γ_a(m); 𝔽_2)  →  H_j(Γ_{a+1}(m); 𝔽_2)
for all 0 ≤ j ≤ h and all m ≥ 1.

Proof.
(a) Stabilize the ring data. Take A_m := t^m k[t] ⊆ k[t]. We must verify
the hypotheses of Iwasa's theorem:
  - Tor-unital: comes from Morrow. ✓
  - sr(A_m) ≤ 2 for every m: direct verification of Vaserstein's
    relative row criterion.
    Fix A-unimodular row b = (b_1, ..., b_r) ∈ (Z ⋉ A_m)^r with r ≥ 3.
    By definition b ≡ e_1 mod t^m over R := k[t]. In particular
    b_1 ≡ 1 mod t^m, so gcd(b_1, t) = 1. For each irreducible divisor
    p of b_1 over k with b_r ≢ 0 mod p, choose v_p ∈ k with
       b_2(p) + v_p b_r(p) ≠ 0
    (possible because there are q > 1 choices of v_p in k = 𝔽_q, at most
    one of which makes b_2(p) + v_p b_r(p) = 0). By Chinese remainder
    theorem on k[t] (a PID), lift the residues v_p to a single
    v ∈ t^m R = A_m.
    Define b' := (b_1, b_2 + v b_r, b_3, ..., b_{r-1}).
    Unimodularity of b': at a divisor p of b_1 where b_r ≢ 0 mod p,
    b_2 + v b_r ≢ 0 mod p by choice of v_p. At a divisor where b_r ≡ 0
    mod p, the original b ∈ (Z ⋉ A_m)^r unimodular (meaning b ≡ e_1 mod
    A_m and b · c = 1 for some c ≡ e_1 mod A_m) allows only finitely
    many prime divisors of b_1; where all of b_2, ..., b_{r-1} vanished
    mod p but b_r did too, one would have b(p) = (0, ..., 0, 0), against
    unimodularity (b · c = 1 mod p means b(p) is not the zero vector).
    Also, at p = (t) (the special divisor of b_r at t): if b_r ≡ 0 mod t
    we already have gcd(b_1, b_r, ..., b_{r-1}) = 1 somewhere else.
    So b' is A-unimodular.
    Lifting back to A-unimodularity over Z ⋉ A_m: given b, c over R
    with b · c = 1 and b ≡ c ≡ e_1 mod t^m, set
      c' := e_1 + (1 - b_1) c.
    Then b · c' = b · e_1 + b · (1 - b_1) c = b_1 + (1 - b_1) b · c = 1,
    and c' ≡ e_1 mod t^m (since 1 - b_1 ≡ 0 mod t^m). So c' gives a
    lift of the unimodularity relation over R to a relation over Z ⋉ A_m:
    the difference between the lifted dot product and 1 lies in A_m,
    where the map to R is injective. So the A-unimodularity criterion is
    satisfied with sr(A_m) ≤ 2.
  - GL_a(A_m) = Γ_a(m). The entrywise map Z ⋉ A_m → R = k[t] is
    injective on A_m. A matrix over Z ⋉ A_m that reduces to one
    congruent to I mod t^m lifts back (the "I + something in A_m" matrix
    comes from a specific element of GL_a(Z ⋉ A_m)). Such a matrix has
    determinant in R^* = k^* that is congruent to 1 mod t^m; but the only
    field element congruent to 1 mod t^m is 1. So det = 1, and the
    matrix lies in SL_a(R) = SL_a(k[t]) ⊆ SL_a(k[t, t^{-1}]). Hence
    GL_a(A_m) = Γ_a(m). ✓

(b) Apply Iwasa. r = 2, so Iwasa gives pro-isomorphisms in INTEGRAL
homology
  (H_j(Γ_a(m); Z))_m → (H_j(Γ_{a+1}(m); Z))_m
is a pro-isomorphism for a ≥ 2j + 1.
Use the universal coefficient theorem
  0 → H_j(G; Z) ⊗ 𝔽_2 → H_j(G; 𝔽_2) → Tor_1^Z(H_{j-1}(G; Z), 𝔽_2) → 0,
natural in G. Applying to the levelwise map Γ_a(m) → Γ_{a+1}(m), and
passing to the pro-category, both Tor and ⊗ extend to endofunctors of
inverse systems and preserve pro-isomorphisms; the levelwise exact
sequences assembled form an exact sequence of inverse systems.
For j ≥ 1, both integral degrees j and j-1 are in the stable range;
degree 0 is trivially equal to 𝔽_2 for all a and m. So we get a
pro-isomorphism:
  (H_j(Γ_a(m); 𝔽_2))_m  →  (H_j(Γ_{a+1}(m); 𝔽_2))_m.
Equivalently, the COKERNEL of the levelwise stabilization map
  H_j(Γ_a(m); 𝔽_2) → H_j(Γ_{a+1}(m); 𝔽_2)
forms a PRO-ZERO inverse system in m.

(c) Upgrade from pro-zero to zero-at-every-level. For m' ≥ m, the index
  [Γ_a(m) : Γ_a(m')]
is finite and odd. To see this, reduce mod t^{m'} and filter matrices
congruent to I mod t^m by successive powers of t; each successive
quotient embeds into an additive matrix group over k (a vector space
over 𝔽_q of size a power of q, hence odd). So the index is a product
of odd numbers, hence odd.
Since the index is odd, we may use the TRANSFER map
  tr: H_j(Γ_a(m); 𝔽_2)  →  H_j(Γ_a(m'); 𝔽_2)
(defined by summing over [Γ_a(m) : Γ_a(m')] coset representatives). The
composition
  inc_* ∘ tr: H_j(Γ_a(m); 𝔽_2) → H_j(Γ_a(m'); 𝔽_2) → H_j(Γ_a(m); 𝔽_2)
equals multiplication by the odd index, which over 𝔽_2 (= characteristic
2 field, but the index is odd integer) is 1 mod 2. So inc_* ∘ tr = 1,
whence inc_* is SURJECTIVE. Equivalently, every transition map
H_j(Γ_a(m'); 𝔽_2) → H_j(Γ_a(m); 𝔽_2) for m' ≥ m is surjective.

(d) Combine (b) and (c). The cokernels of the levelwise stabilization
maps H_j(Γ_a(m)) → H_j(Γ_{a+1}(m)) form an inverse system whose
transition maps are SURJECTIVE (from (c)) AND whose limit is pro-zero
(from (b)). A pro-zero inverse system with surjective transition maps
is zero at every level: given m, choose m' ≥ m with the transition
m' → m zero (pro-zero); but the same transition is surjective by (c),
so its target H is both the image of the zero map (so {0}) and the
whole of H (so H itself). So H = 0. Therefore the cokernels vanish
levelwise: H_j(Γ_a(m); 𝔽_2) → H_j(Γ_{a+1}(m); 𝔽_2) is surjective for
every m ≥ 1, in the stated range a ≥ 2j + 1 for 0 ≤ j ≤ h. ∎

Part 3 (continued). Trivial conjugation and vanishing cohomology.

Exercise: Lemma (Trivial Conjugation). State and prove.
For 1 <= j <= d - 1, the conjugation action of SL_N(R) on
H_j(Γ_N(m); 𝔽_2) is trivial, for every m >= 1.

Proof.
(1) Stability chain. Apply the Stabilization lemma twice, at h = d - 1.
  - First with a = N - 2 = 2d - 1: hypothesis a >= 2h + 1 = 2(d-1) + 1 =
    2d - 1 is met. So H_j(Γ_{N-2}(m); F_2) → H_j(Γ_{N-1}(m); F_2) is
    surjective for 0 <= j <= d - 1.
  - Then with a = N - 1 = 2d: hypothesis a >= 2d - 1 is met. So
    H_j(Γ_{N-1}(m); F_2) → H_j(Γ_N(m); F_2) is surjective for j <= d - 1.
  Composing: the natural map
    H_j(Γ_{N-2}(m); F_2) → H_j(Γ_N(m); F_2)
  is surjective for 1 <= j <= d - 1. Here Γ_{N-2}(m) sits on the first
  N-2 coordinates, fixing the last 2 as the identity.

(2) Same conclusion after permuting coordinates. The signed permutation
  group Sym(N) x {+-1}^N intersects SL_N(R) as a subgroup of signed
  permutation matrices (with determinant 1). Any such group element g
  normalizes Γ_N(m), so it induces a homology automorphism. Consequently
  the surjectivity from (1) also holds for the subgroup
    Γ_T(m) := {(M)_{a,b in T}, identity elsewhere} ⊆ Γ_N(m)
  where T is ANY (N - 2)-subset of S, provided T ranges over all
  (N-2)-subsets (conjugates of the first N-2 by signed permutations).

(3) Elementary matrices centralize complementary congruence subgroups.
  Fix distinct a, b in S, and let T := S \ {a, b}. Then Γ_T(m) fixes
  coordinates a and b. The elementary matrix e_ab(f) = I + f E_{a,b}
  acts by a matrix that is the identity on all coordinates except a
  and b. Conjugating Γ_T(m) by e_ab(f) leaves it pointwise fixed:
    e_ab(f) Γ_T(m) e_ab(f)^{-1} = Γ_T(m).
  Therefore e_ab(f) acts trivially on H_j(Γ_T(m); F_2) for such T.

(4) Combine. From (2), the image of H_j(Γ_T(m)) → H_j(Γ_N(m)) is all of
  H_j(Γ_N(m)) (surjection). From (3), e_ab(f) acts trivially on the
  source of this surjection. So e_ab(f) acts trivially on its image, which
  is all of H_j(Γ_N(m)).

(5) General SL_N(R). The elementary matrices generate SL_N(R) when R is
  a Euclidean domain (standard argument: Euclidean ⇒ elementary
  operations reduce any matrix in SL_n(R) to identity). So the whole
  group SL_N(R) acts trivially on H_j(Γ_N(m); F_2) for 1 <= j <= d-1.
  ∎

Exercise: Proposition (Vanishing Cohomology of T_m). State and prove.
For every m >= 2,
  H_j(Γ_N(m); 𝔽_2) = H_j(T_m; 𝔽_2) = H^j(T_m; 𝔽_2) = 0
for 1 <= j <= d - 1.

Proof.
(1) |G_m| is odd. Filter the group G_m = U / Γ_N(m) by congruence levels:
    U  =  Γ_N(0) ⊇ Γ_N(1) ⊇ Γ_N(2) ⊇ ... ⊇ Γ_N(m).
  Each layer Γ_N(i+1) / Γ_N(i) consists of matrices of the form I + M t^i
  modulo t^{i+1}, where M ranges over an additive subgroup of N×N matrices
  over k = 𝔽_q (with structural zeros where U imposes zero; upper triangular
  constraints at i = 0). Its order is q^{dim}, hence ODD since q is odd.
  Each quotient Γ_N(i) / Γ_N(i+1) thus has odd order. Also U / Γ_N(1) ⊆
  the upper unitriangular group of order q^{N(N-1)/2}, odd. So |G_m| is a
  product of odd factors, hence odd.

(2) Pick a projective 𝔽_2[U]-resolution P_* → 𝔽_2 of the trivial module
  𝔽_2.
  Restriction to Γ_N(m): 𝔽_2[U] splits as a direct sum of translates of
  𝔽_2[Γ_N(m)] (one per left coset of Γ_N(m) in U), so 𝔽_2[U] is a FREE
  𝔽_2[Γ_N(m)]-module. Hence each P_j is projective over 𝔽_2[Γ_N(m)], and
  (P_*)_{Γ_N(m)} := P_* ⊗_{𝔽_2[Γ_N(m)]} 𝔽_2 computes
  H_j(Γ_N(m); 𝔽_2).

(3) G_m acts. Since Γ_N(m) is normal in U, the quotient G_m acts by left
  multiplication on F_2[U], hence on the complex (P_*)_{Γ_N(m)}, and on
  the homology H_j(Γ_N(m); F_2). This is the CONJUGATION action induced
  by the conjugation action of U on Γ_N(m) = ker(U → G_m).

(4) Coinvariants via odd-order averaging. The coinvariant functor
  M ↦ M_{G_m} := M ⊗_{F_2[G_m]} F_2 is exact because |G_m| is odd: the
  augmentation F_2[G_m] → F_2 splits (multiplication by |G_m|^{-1}, which
  is 1 in F_2 as |G_m| is odd). So:
    H_j(Γ_N(m); F_2)_{G_m}
      = H_j(((P_*)_{Γ_N(m)})_{G_m})
      = H_j((P_*)_{U})       [via  Γ_N(m) ⋊ G_m = U,
                              hence M_U = (M_{Γ_N(m)})_{G_m}]
      = H_j(U; F_2).
  (The second equality uses the standard identity
  (M)_{Γ_N(m) then G_m} = (M)_U for M a F_2[U]-module, valid because
  F_2[U] = F_2[Γ_N(m)] ⊗_{F_2} F_2[G_m] is a tensor product.)

(5) Trivial action. By the Trivial Conjugation lemma, for 1 <= j <= d-1,
  G_m ⊆ SL_N(R) acts trivially on H_j(Γ_N(m); F_2). Therefore the
  coinvariants equal the module itself:
    H_j(Γ_N(m); F_2)_{G_m} = H_j(Γ_N(m); F_2).

(6) Combine (4) and (5). For 1 <= j <= d-1,
    H_j(Γ_N(m); F_2) = H_j(U; F_2) = 0   (by Prop from Part 3).
  By the quotient Prop (H_j(T_m) = H_j(Γ_N(m)) for 0 < j < n-1), and since
  d-1 < n-1, we get H_j(T_m; F_2) = 0 for 1 <= j <= d-1.

(7) Cohomology. By the universal coefficient theorem over 𝔽_2, for any
  finite simplicial complex Y and any j >= 0,
    0 → Ext^1(H_{j-1}(Y), 𝔽_2) → H^j(Y; 𝔽_2) → Hom(H_j(Y), 𝔽_2) → 0.
  Since 𝔽_2 is a field, every 𝔽_2-vector space is both projective and
  injective, so Ext^1 = 0 for all coefficients in 𝔽_2. Thus
  H^j(Y; 𝔽_2) = Hom(H_j(Y; 𝔽_2), 𝔽_2).
  For Y = T_m and 1 <= j <= d-1: H_j(T_m; F_2) = 0 by (6), so
  Hom(H_j(T_m; F_2), F_2) = 0. Hence H^j(T_m; F_2) = 0. ∎

Exercise: What does vanishing of H^j(T_m) give us?
For the cochain complex of T_m over F_2:
  C^0 → C^1 → C^2 → ... → C^n
with coboundaries B^i = im δ_{i-1} ⊆ C^i and cocycles Z^i = ker δ_i ⊆ C^i,
we have H^i = Z^i / B^i. Vanishing H^i(T_m; F_2) = 0 for
1 <= i <= d-1 says B^i = Z^i exactly: every cocycle is a coboundary.
Therefore
  dist(f, Z^i(T_m)) = dist(f, B^i(T_m)) for all f ∈ C^i, 1 <= i <= d-1.
In particular, any quantitative cosystolic inequality
  ||δ_i f|| >= η · dist(f, Z^i(T_m)), 1 <= i <= d-1,
becomes automatically a coboundary expansion inequality
  ||δ_i f|| >= η · dist(f, B^i(T_m)), 1 <= i <= d-1.
That is the entire role of Step 3 in the proof: to replace cocycles
by coboundaries in the KMS cosystolic inequality from Step 4.

Part 4. The cosystolic input from KMS complexes.

The remaining steps are: use a KNOWN theorem (from the KMS
higher-dimensional solver theory) to get expansion against cocycles,
then do the skeleton-weight bookkeeping to push it down to the
d-skeleton of X_m.

Exercise: What is a KMS complex?
The affine Coxeter diagram Ã_n has nodes S = {0, ..., n} forming a
cycle (S has n+1 = N nodes). A "KMS" structure is a specific way of
building a group out of the subdiagrams:
  - For each PROPER J ⊊ S, associate a finite group U_J with inclusions
    U_J → U_K for J ⊆ K.
  - The "universal completion" Û is the free product of all these U_J
    modulo the identification U_J = image in U_K.
  - A finite quotient φ: Û ↠ G defines a "KMS complex" when φ is
    injective on every spherical local group U_J and preserves all
    intersections. The resulting complex is CC(G; (φ(U_{S \ {i}}))_{i}).
A diagram is called n-classical when every proper subdiagram has
irreducible components of classical type A_r, B_r, C_r, D_r. The
affine Ã_n diagram is n-classical because deleting any node turns the
cycle into a path of type A.

Exercise: Theorem (OVB cosystolic expansion). State the input.
Oppenheim-Valentiner-Branth [OVB], revised September 2026, arXiv:2504.05823v2,
to appear in Math. Ann. Their Theorem 1.2 (specialized to our needs):
Fix n >= 3, an n-classical diagram on S with |S| = n+1, and a
sufficiently large prime q. For every finite n-dimensional n-classical
KMS complex Y over 𝔽_q, there is a constant η = η(n, S, q) > 0,
independent of the finite quotient defining Y, with
  ||δ_i f||_{Y, i+1}  >=  η · dist_{Y, i}(f, Z^i(Y))
     for all 0 <= i < n - 1 and all f ∈ C^i(Y; 𝔽_q).
The weights are exactly those of eq. (weights) above.
We use this in the special case Y = T_m at the (n-1)-skeleton.
Important: this is expansion against COCYCLES (distance to Z^i), not
coboundaries. Part 3 fills the gap by showing Z^i = B^i for 1 <= i <= d-1.

Exercise: Verify T_m satisfies all the KMS hypotheses.
(a) Dimension and classicalness. T_m has dimension N-1 = n. Every proper
  subdiagram of Ã_n is a path of type A_{|J|-1}, which is A_r. So T_m is
  n-classical.
(b) Realization of the universal completion. For proper J ⊆ S, define
  U_J := <x_j : j ∈ J> ⊆ U. Lemma (Local Groups) describes U_J as a
  finite unitriangular-pattern subgroup. The inclusions U_J ⊆ U_K for
  J ⊆ K are homomorphisms, and their images inside U agree with the
  diagram-prescribed inclusions. So the universal completion Û has a
  natural map to U, which is SURJECTIVE by Generation Corollary.
(c) Local injectivity and intersections. By Lemma (Local Groups), the
  composite Û → U → G_m = U / Γ_N(m) is injective on every spherical
  local group U_J (m >= 2), and preserves all pairwise intersections
  (U_J ∩ U_K = U_{J ∩ K}). So T_m = CC(G_m; (H_i)_{i in S}) is a finite
  n-dimensional n-classical KMS complex.
(d) The weights of T_m match eq. (weights) for the theorem. So the
  cosystolic inequality on the (n-1)-skeleton Y_m = skel_{n-1} T_m holds:
    ||δ_i f||_{Y_m, i+1}  >=  η · dist_{Y_m, i}(f, Z^i(Y_m))
       for 0 <= i < n - 1,
  with η > 0 depending only on n, q, and S (not on m).

Exercise: Choose q once and for all.
The requirements on q accumulated so far:
  (1) q odd prime (Lemma Local Groups).
  (2) q >= 5 (to avoid small-characteristic degeneracies).
  (3) q >= 2^{n-1} (spherical opposition vanishing, Part 2).
  (4) q >= threshold from OVB Theorem 1.2 (finite list, only bounded by n).
Each is a finite lower bound on q, depending only on n. Choose a prime q
satisfying all of them. Since there are arbitrarily large primes, this
is possible. From here on q is FIXED and all constants (M, D, η, ε)
are independent of m and of i.

Exercise: Degree bound. Show that every vertex of X_m lies in at most
D = M * binom(N - 1, d) faces of dimension d (with M := q^{binom(N,2)}
and D independent of m).
Proof. The number of chambers of T_m through any vertex v is exactly M.
Reason: chambers of T_m are in bijection with elements of G_m (the
common stabilizer of the base-chamber vertices is U_empty = {1}); the
stabilizer of the type-i vertex of the base chamber is H_i of size M.
So there are exactly M chambers through the base type-i vertex, and
by G_m-transitivity M through any type-i vertex.
A d-face through v = a subset of d+1 vertices containing v. Every
d-face through v lies inside SOME chamber through v. So the number of
d-faces through v is at most (number of chambers through v) × (number
of (d+1)-subsets of a chamber containing v as first element)
   = M · binom(N - 1, d)  =:  D.
(D is an upper bound, not a tight count. That is enough for the theorem.)
Since D does not involve m, the "bounded vertex degree" condition (a) of
the main theorem holds. ∎

Exercise: Growth. Show |X_m(0)| → ∞.
The total number of chambers of T_m is |G_m|. Each type i ∈ S appears
in exactly |G_m| / |H_i| = |G_m| / M chambers (by transitivity and
the stabilizer sizes). So the total number of type-i vertices is
|G_m| / M, and
   |X_m(0)| = N · |G_m| / M.
Now |G_m| ≥ q^m: the subgroup {e_{12}(f) : f ∈ k[t]} ⊆ U has image in
G_m of size at least q^m (polynomials of degree < m give distinct
images mod t^m). Hence
   |X_m(0)| >= N · q^m / M → ∞   as m → ∞,
with the rate of divergence controlled only by n and M (not by any
other choice). ∎

Part 5. Weight comparison and the main theorem.

Exercise: How do weights change when we pass to a skeleton?
Let T be a finite pure n-complex, X := skel_s(T) for some 0 <= s <= n.
Write μ_i := w_i^T and ν_i := w_i^X for the intrinsic weights of i-faces
in T and in X respectively. For each s-face τ ∈ T(s), let c(τ) be the
number of n-faces (chambers) of T containing τ. Suppose
  0 < a <= c(τ) <= b       for all τ ∈ T(s).
Define a constant (depending only on T and s, not on i or σ)
  Λ := binom(n+1, s+1) · |T(n)| / |T(s)|.
Then for every i-face σ with 0 <= i <= s,
  (Λ/b) · μ_i(σ)  <=  ν_i(σ)  <=  (Λ/a) · μ_i(σ).

Proof. For i = s: by the weight definition,
  μ_s(σ) = c(σ) / (binom(n+1, s+1) · |T(n)|)
  ν_s(σ) = 1 / |T(s)|
So ν_s(σ) / μ_s(σ) = Λ / c(σ), and the bounds follow directly.
For i < s: use the two averaging identities
  μ_i(σ) = binom(s+1, i+1)^{-1} · sum_{τ ∈ T(s), σ ⊆ τ} μ_s(τ)
  ν_i(σ) = binom(s+1, i+1)^{-1} · sum_{τ ∈ T(s), σ ⊆ τ} ν_s(τ).
The first is a double count of (chamber, face) incidences in T; the
second is the definition of skeleton weights on X (which is the s-skeleton
of T, so its s-faces are T(s)). Sum the i = s bounds over τ ⊇ σ:
  (Λ/b) sum μ_s(τ) <= sum ν_s(τ) <= (Λ/a) sum μ_s(τ),
and divide by binom(s+1, i+1). Same inequalities hold for norms and for
distances to any subspace (all comparisons are faces-wise and positive).∎

Exercise: Apply the weight-comparison lemma twice to T_m.
(1) T = T_m, X = Y_m := skel_{n-1}(T_m), s = n-1.
Every (n-1)-face (panel) of T_m has |U_{S\A}| = |U_one| = q chambers. So
a = b = q. The constant Λ = binom(n+1, n) |T_m(n)| / |T_m(n-1)|.
Double-counting (chamber, panel) incidences in T_m:
  (n+1) · |T_m(n)|  =  number of (chamber, panel inside it) pairs
  q · |T_m(n-1)|    =  same number,
so Λ = q. Thus Λ/a = Λ/b = 1 and the weights coincide in common degrees:
  ||f||_{Y_m, i} = ||f||_{T_m, i}     for 0 <= i <= n-1,
and likewise for any distance to a subspace.

(2) T = T_m, X = X_m := skel_d(T_m), s = d.
Every d-face (type A ⊆ S with |A| = d+1) is contained in between 1 and
M chambers (lower bound: purity gives at least 1; upper bound: chambers
through a face of type A = |U_{S\A}| <= |H_0| = M). So a = 1, b = M.
The constant Λ_m := binom(N, d+1) |T_m(n)| / |T_m(d)|. Then for every
cochain f and 0 <= i <= d:
  (Λ_m/M) ||f||_{Y_m, i} <= ||f||_{X_m, i} <= Λ_m ||f||_{Y_m, i},
and for every subspace A ⊆ C^i(Y_m) = C^i(X_m):
  (Λ_m/M) dist_{Y_m, i}(f, A) <= dist_{X_m, i}(f, A) <= Λ_m dist_{Y_m, i}(f, A).

The constant Λ_m is common to all degrees 0 <= i <= d, and will cancel
in the final expansion inequality.

Exercise: Prove the main theorem.
Let ε := η / M and D := M · binom(N-1, d), with η = η(n, S, q) from OVB
Theorem 1.2 and M = q^{binom(N,2)}. Define X_m := skel_d(T_m) for m >= 2.

Step A: The complexes X_m satisfy all structural hypotheses.
Finiteness: each T_m is a finite complex (Lemma Local Groups). Purity:
both T_m and X_m = skel_d(T_m) are pure. Connectedness: T_m is connected
(Part 3), so X_m is connected. Degree bound: D above. Growth:
|X_m(0)| → ∞ (Part 4).

Step B: Cohomology of T_m vanishes in the range 1 <= j <= d-1.
By Part 3 Proposition (Vanishing Cohomology),
  H^j(T_m; F_2) = 0 for 1 <= j <= d-1.
Consequently, in the cochain complex of T_m over F_2,
  Z^j(T_m) = B^j(T_m) for 1 <= j <= d-1.
When j = 0, H^0(T_m) = F_2 (T_m is connected), and Z^0 = B^0 = constants.
So in fact Z^j(T_m) = B^j(T_m) for 0 <= j <= d-1.

Since X_m = skel_d(T_m) and Y_m = skel_{n-1}(T_m), and d <= n-1, we have
C^j(X_m) = C^j(Y_m) = C^j(T_m) for 0 <= j <= d (they see the same
j-faces), and the coboundary map δ_j is the same. So
  Z^j(Y_m) = B^j(X_m) for 0 <= j <= d-1. Denote this common subspace V_j.

Step C: Fix m >= 2, 0 <= i < d, f ∈ C^i(X_m; F_2). Compute.
  ||δ_i f||_{X_m, i+1}
    >= (Λ_m/M) · ||δ_i f||_{Y_m, i+1}           [weight comparison, degree i+1]
    >= (Λ_m/M) · η · dist_{Y_m, i}(f, Z^i(Y_m)) [OVB cosystolic on Y_m]
    =  (Λ_m/M) · η · dist_{Y_m, i}(f, B^i(X_m)) [Z^i(Y_m) = B^i(X_m)]
    >= (Λ_m/M) · η · (1/Λ_m) · dist_{X_m, i}(f, B^i(X_m))
                                                 [distance comparison, degree i]
    =  (η/M) · dist_{X_m, i}(f, B^i(X_m))
    =  ε · dist_{X_m, i}(f, B^i(X_m)).
Here Λ_m cancels between the two comparison inequalities. So the expansion
holds with ε = η/M > 0, independent of m and i.

Step D: Conclude. The five conditions of the main theorem are:
  (a) X_m finite, connected, pure of dimension d. ✓
  (b) |X_m(0)| → ∞. ✓
  (c) Every vertex in at most D faces of dimension d. ✓
  (d) Uniform coboundary expansion ||δ_i f|| >= ε·dist(f, B^i) for
      0 <= i < d with ε = η/M independent of m and i. ✓
  (e) D < ∞ and ε > 0 independent of m. ✓
The constants D and ε depend only on n, S, q. Since q was fixed once and
for all at the start of Part 4, D and ε are uniform. So the main theorem
is proven for all d >= 3. ∎

Exercise: Why does the theorem require d >= 3?
Our construction works for d = 1 and d = 2 as well, but those are already
known:
- d = 1: expander graphs (Lubotzky 1994, and predecessors).
- d = 2: Chapman-Lubotzky 2025 (two-dimensional F_2 coboundary expanders),
  also via KOW2025 methods.
For d >= 3, the theorem is new. Together with the known d = 1 and d = 2
cases, the construction gives bounded-degree F_2 coboundary expanders in
every positive dimension, which is the abstract's conclusion.
The reason we require d >= 3 specifically in THIS proof is that the OVB
cosystolic theorem (Step 4) is stated for n >= 3 (n = 2d, so d >= 2),
plus some small-prime technicalities that only matter when the dimension
is 1 or 2, but are comfortably absorbed by d >= 3. So d >= 3 is a
"comfortable range" assumption, not a fundamental barrier.

Part 6. Where to learn more.

Exercise: Where to learn more (key references, in reading order).
Background on expander graphs and higher-dimensional expansion:
  Gromov 2010, GAFA 20 (higher-dimensional filling problem).
    doi:10.1007/s00039-010-0073-8
  Lubotzky 1994, Discrete Groups, Expanding Graphs and Invariant Measures.
    doi:10.1007/978-3-0346-0398-0

The KMS input (the main quantitative step):
  Grave de Peralta - Valentiner-Branth 2025, EJC 126.
    doi:10.1016/j.ejc.2025.104131
  Oppenheim - Valentiner-Branth 2026, Math. Ann. (to appear).
    arxiv:2504.05823v2 -- Theorem 1.2 is our cosystolic input.

Buildings and CAT(0) geometry (used in Part 2):
  Abramenko - Brown 2008, GTM 248.
    doi:10.1007/978-0-387-78835-7
  Abramenko - Van Maldeghem 2001, Math. Z. 238 (1-twinnings).
    doi:10.1007/PL00004899
  Bridson - Haefliger 1999, Metric Spaces of Non-Positive Curvature.
    doi:10.1007/978-3-662-12494-9

Stability theory (used in Part 3):
  Vaserstein 1969, Math. USSR-Sb. 8.
    doi:10.1070/SM1969v008n03ABEH001279
  Morrow 2018, J. reine angew. Math. 736 (Tor-unitality).
    arxiv:1404.4179v2
  Iwasa 2020, Homology Homotopy Appl. 22 (pro-stability).
    doi:10.4310/HHA.2020.v22.n1.a20

Earlier work on bounded-degree expansion, and the d = 1, d = 2 cases:
  Linial - Meshulam 2006, Combinatorica 26.
  Lubotzky - Luria - Rosenthal 2019, DCG 62.
  Kaufman - Kazhdan - Lubotzky 2016, GAFA 26.
  Evra - Kaufman 2024, J. Amer. Math. Soc. 37.
  Dikstein - Dinur 2024, APPROX/RANDOM 2024.
  Kaufman - Oppenheim 2023, EJC 111.
  Kaufman - Oppenheim - Weinberger 2025, STOC 2025.
  Chapman - Lubotzky 2025, Adv. Math. 463.

End of study guide.