Prior work.

The higher-dimensional coisoperimetric viewpoint developed through the random-complex work of Linial and Meshulam [16] and Meshulam and Wallach [20]. Their models retain a complete lower skeleton and study the disappearance of cohomology as top-dimensional faces are added. Gromov connected cohomological filling inequalities with topological overlap and asked for arbitrarily large complexes with bounded vertex incidence and uniform filling bounds [9]. The full coboundary expansion problem also demands vanishing cohomology; a recent formulation is [5].

There are two different degree conditions in this history. Lubotzky and Meshulam constructed random Latin-square coboundary expanders in dimension two [18]; Lubotzky, Luria, and Rosenthal constructed random Steiner-system coboundary expanders in every dimension [17]. These results bound the number of top-dimensional faces through a codimension-one face. The Latin-square complexes have a complete tripartite \(1\)-skeleton, and the Steiner-system complexes have a complete \((d-1)\)-skeleton. In dimensions at least two, vertex degrees grow with the size. The degree bound in 1 is at vertices.

For bounded vertex degree, Kaufman, Kazhdan, and Lubotzky established cosystolic and topological expansion for two-dimensional skeleta of Ramanujan complexes [11]. Evra and Kaufman obtained bounded-degree cosystolic expanders in every dimension through a local-to-global theorem [7]. The associated expansion methods include the building-like complexes of Lubotzky, Meshulam, and Mozes [19], the strongly symmetric complexes of Kaufman and Oppenheim [13], the coefficient-independent methods of Kaufman and Mass [12], and the quantitative local-to-global results of Dikstein and Dinur [6]. We use only the distance-to-cocycles part of cosystolic expansion, stated precisely in (16).

Chapman and Lubotzky proved the existence of bounded-degree two-dimensional coboundary expanders over \(\mathbb F_2\) [5]. In higher ambient dimensions, Kaufman, Oppenheim, and Weinberger [15], and Oppenheim and Valentiner-Branth [24], proved degree-one coboundary expansion for coset complexes with suitable non-Abelian coefficients. Those degree-one results and the cosystolic constructions above do not give the vanishing in every degree required by 1. Together with the two-dimensional result and the graph case, our theorem gives bounded-degree \(\mathbb F_2\) coboundary expanders in every positive dimension.

Our complexes belong to an established algebraic construction program. Kaufman and Oppenheim developed locally spectral expanding coset geometries [14]; O’Donnell and Pratt used Chevalley groups [22]; and Grave de Peralta and Valentiner-Branth developed Kac–Moody–Steinberg group constructions [8]. Oppenheim and Valentiner-Branth proved the cosystolic expansion theorem for these KMS complexes that supplies our quantitative input [23]. Their affine-type-\(A\) polynomial matrix construction [23] already uses the simple root groups appearing here, with quotients by irreducible polynomials. Powers of \(t\) also occur, with different local groups, in the degree-one construction of [15]. For the local groups used here, we pass to the tower of quotients modulo \(t^m\) and prove the cohomology vanishing needed to obtain coboundary expansion in all degrees below \(d\).

Proof strategy.

Fix \(d\) and put \(n=2d\) and \(N=2d+1\). Over a sufficiently large fixed odd prime field \(k\), let \(U\le\operatorname{SL}_N(k[t])\) be the inverse image of the upper unitriangular group under evaluation at zero. We construct an infinite \(n\)-dimensional coset complex \(L\) for \(U\) and its finite congruence quotients \(T_m\). The desired \(X_m\) is the \(d\)-skeleton of \(T_m\). The local groups are fixed finite groups, which gives uniform degree bounds; the congruence quotients grow with \(m\).

The first step is geometric. We identify \(L\) with the subcomplex opposite a chamber in an affine twin building. A filtration by codistance expresses the relative chain groups of each attachment in terms of a spherical opposition complex. The opposition-complex input of [23] makes these attachments acyclic in the required range. Since the whole affine building is contractible, \(L\) has vanishing reduced homology below degree \(n-1\). Its simplex stabilizers have odd order, so its chain complex gives a partial projective resolution over \(\mathbb F_2[U]\). The orbit complex \(U\backslash L\) is a single simplex, which yields the corresponding homology vanishing for \(U\).

The second step passes from \(U\) to each congruence subgroup. Morrow’s pro Tor-unitality theorem [21] and Iwasa’s homology pro-stability theorem [10] apply to the ideals \(t^m k[t]\). Their initial conclusion concerns an inverse system, whereas we need a statement at every level. Odd-index transfer makes the homology maps towards shallower levels surjective over \(\mathbb F_2\). The cokernels of stabilization are therefore both pro-zero and connected by surjective transition maps, so each cokernel vanishes. Two stabilization steps then show that ambient conjugation acts trivially in degrees \(1,\ldots,d-1\). Averaging over the finite odd-order quotient transfers the vanishing from \(U\) to the congruence subgroup, and hence to \(T_m\).

Finally, the KMS theorem gives a uniform inequality against cocycles on the \((n-1)\)-skeleton of \(T_m\). Its weighted norms agree with the corresponding norms on \(T_m\) because every panel lies in the same number of chambers. Passing to the \(d\)-skeleton requires a further comparison: the number of chambers through a \(d\)-face is bounded above and below, and the normalization contributes the same factor in two successive cochain degrees. This factor cancels in the expansion inequality. The cohomology vanishing replaces cocycles by coboundaries and completes the proof.

Organization.

Section 2 defines the congruence complexes. Section 3 proves the local group identities and the acyclicity of \(L\). Section 4 proves the cohomology vanishing of the finite quotients. Section 5 checks the KMS hypotheses, compares skeleton weights, and proves 1.

The finite local groups

Regard \(S\) as the edges of the directed cycle \(1\to2\to\cdots\to N\to1\), with edge \(0\) equal to \(N\to1\). For \(J\subsetneq S\), put \(U_J=\langle x_j:j\in J\rangle\) and \(U_{\varnothing}=\{1\}\). Between distinct indices \(a,b\) there is a unique simple directed path in the cycle. Set \[V_{ab}(J)= \begin{cases} k,&\text{if that path lies in $J$ and avoids edge $0$},\\ kt,&\text{if that path lies in $J$ and uses edge $0$},\\ 0,&\text{otherwise}. \end{cases}\]

The twin-building model

The matrix calculation identifies the stabilizers we need. To obtain acyclicity, we now place the coset complex in the classical affine twin building of \(\operatorname{SL}_N(k[t,t^{-1}])\). We use the lattice construction of Abramenko–Van Maldeghem [3]; see also [1] and [2]. Its two halves are the affine buildings at \(0\) and at infinity, both of type \(\widetilde A_n\) and dimension \(n\). Their chambers have one vertex of each type in \(S\). The stabilizers of opposite base chambers are \[B_+=\{g\in\operatorname{SL}_N(k[t]):g(0)\text{ is upper triangular}\},\] and the analogous group \(B_-\) over \(k[t^{-1}]\) with lower triangular reduction at infinity. They satisfy \(B_+=U\rtimes T\) and \(B_+\cap B_-=T\), where \(T\) is the constant diagonal torus.

Fix the positive base chamber \(c_+\). A chamber \(c\) in the negative building has a codistance \(\delta^*(c)=\delta^*(c_+,c)\in W\), where \((W,S)\) is the affine Coxeter system. For two chambers in the same half, their Weyl distance is the element of \(W\) represented by the types of a minimal gallery between them. Write \(\ell\) for word length with respect to \(S\), and \(W_J=\langle J\rangle\) for \(J\subseteq S\). The chambers of codistance \(1\) are exactly the \(B_+\)-orbit of the negative base chamber. We use the convention that an \(s\)-panel varies the vertex of type \(s\). If \(\delta^*(c)=w\), the codistances in this panel belong to \(\{w,ws\}\); the value \(ws\) is attained, and, when \(\ell(ws)<\ell(w)\), every chamber in the panel other than \(c\) has codistance \(ws\). These are the twin-building axioms with the positive argument fixed [3]. Every proper \(J\subsetneq S\) is spherical: a \(J\)-residue, obtained by varying only types in \(J\), is a finite spherical building whose irreducible factors have type \(A\) over \(k\). In the lattice model, fixing the other types fixes a partial lattice chain; completing that chain amounts to choosing flags in its successive finite-dimensional \(k\)-vector-space quotients. This gives the asserted type-\(A\) factors.

Let \(L'\) be the subcomplex consisting of all codistance-\(1\) chambers and their faces. The preceding description shows that \(U\) acts simply transitively on its chambers. We check their vertex stabilizers. In the standard lattice model at infinity, vertices are homothety classes of \(k[[t^{-1}]]\)-lattices. Choose the negative base chamber with vertices represented by exponent vectors \[(0,\ldots,0)\quad\text{(type $0$)},\qquad (\underbrace{0,\ldots,0}_{i},\underbrace{1,\ldots,1}_{N-i}) \quad\text{(type $i$)}.\] For a vertex represented by exponents \((b_a)\), a polynomial matrix in its stabilizer satisfies \[ \deg g_{ab}\le b_a-b_b. \tag{9}\] This is the condition for preserving the lattice generated by \(t^{b_a}e_a\) over \(k[[t^{-1}]]\); because the determinant is \(1\), preservation of its homothety class implies preservation of the lattice itself. Intersecting these bounds with \(U\) gives upper unitriangular constant matrices at the type-\(0\) vertex. At type \(i\ne0\) they give two upper unitriangular constant diagonal blocks, cut after coordinate \(i\), a zero upper-right block, and arbitrary entries in \(kt\) in the lower-left block. By 1, these stabilizers are exactly \(H_0,H_i\), respectively. Sending \(uH_i\) to the corresponding vertex identifies \[ L=\operatorname{CC}(U;(H_i))\cong L'. \tag{10}\] This uses the common-representative definition of faces: a face of \(L'\) lies in one of its chambers, and every such chamber is a unique translate of the base chamber.

A filtration by codistance

We have identified \(L\) with an opposition subcomplex. The next step is to compare it with the whole negative building by adding chambers in codistance order. The homology of each attachment is controlled by spherical opposition complexes, for which we use the following input. In a spherical building of type \(A_r\) over \(k\), let \(O\) be the subcomplex formed by chambers opposite a fixed chamber and their faces. For \(q\ge2^{r-1}\), \[ \widetilde H_j(O;\mathbb F_2)=0\qquad(j<r-1). \tag{11}\] This follows from the integral cone function in [23], whose chain contraction identity remains valid after reduction modulo two. For \(r=1\), the required assertion is just nonemptiness of the opposition complex. For a product of spherical buildings, the opposition complex is the simplicial join of the opposition complexes of the factors, and the augmented chain formula for joins gives the same vanishing below its dimension. We henceforth require \(q\ge2^{n-1}\), which covers every proper residue of our affine building.

An acyclic complex as a partial resolution

Finite odd-order stabilizers make the simplicial chains projective over the group algebra. This allows the acyclicity already proved for \(L\) to replace a full contractible model in the degrees we need.

From pro-stability to ordinary surjectivity

Our next goal is a surjection on homology at each fixed congruence level. The available stability theorem concerns inverse systems; odd-index transfer will supply the additional step from that theorem to an ordinary surjection.

We first specify its ring conventions. For a commutative nonunital ring \(A\), its integer unitization is \(\mathbb Z\ltimes A=\mathbb Z\oplus A\), with \[(u,a)(v,b)=(uv,ub+va+ab),\qquad 1=(1,0).\] The projection to \(\mathbb Z\) defines \[\operatorname{GL}_a(A)=\ker\bigl(\operatorname{GL}_a(\mathbb Z\ltimes A)\longrightarrow\operatorname{GL}_a(\mathbb Z)\bigr).\] A row \(b\in(\mathbb Z\ltimes A)^\ell\) is \(A\)-unimodular if \(b\equiv e_1\pmod A\) and there is a row \(c\equiv e_1\pmod A\) with \(b\cdot c=1\). Here \(e_1=(1,0,\ldots,0)\). We use the relative stable range convention of Vaserstein [25]: \(\operatorname{sr}(A)\le s\) means that, for every such row of length \(\ell\ge s+1\), there are \(v_1,\ldots,v_{\ell-1}\in A\) for which \[(b_1+v_1b_\ell,\ldots,b_{\ell-1}+v_{\ell-1}b_\ell)\] is \(A\)-unimodular.

An inverse system \((A_m)\) is pro-zero if, for every index, some later transition to that index is zero. A level map of inverse systems of abelian groups is a pro-isomorphism precisely when its kernel and cokernel systems are pro-zero. The system \((A_m)\) is Tor-unital if \((\operatorname{Tor}_i^{\mathbb Z\ltimes A_m}(\mathbb Z,\mathbb Z))_m\) is pro-zero for every \(i>0\). Morrow’s theorem says that the powers of any ideal in a commutative Noetherian ring form a Tor-unital system [21]. For a Tor-unital inverse system of commutative nonunital rings, Iwasa’s integral homology pro-stability theorem says that, if \(r=\max\{2,\sup_m\operatorname{sr}(A_m)\}<\infty\), then \[ (H_j(\operatorname{GL}_a(A_m);\mathbb Z))_m\longrightarrow (H_j(\operatorname{GL}_{a+1}(A_m);\mathbb Z))_m \quad\text{is a pro-isomorphism for }a\ge2j+r-1. \tag{15}\] We cite the published version [10].

Trivial conjugation and vanishing cohomology

Stabilization lets us represent every homology class on any chosen set of \(N-2\) coordinates. The two remaining coordinates will make each elementary conjugation act trivially. This is the finite-level argument corresponding to the stability-to-conjugation principle in [10].

We can now recover all the required congruence-group homology from its coinvariants under \(G_m\). The odd order of \(G_m\) makes taking these coinvariants exact, and the preceding lemma makes their action trivial.

The quantitative input and its hypotheses

We recall the Kac–Moody–Steinberg (KMS) construction in the case needed here. The affine diagram \(\widetilde A_n\) has type set \(S\), and each proper subdiagram is a disjoint union of paths of type \(A\). Its spherical local groups are the corresponding unitriangular groups \(U_J\), \(J\subsetneq S\), with their standard inclusions. Their universal completion is the group obtained by taking the free product of these groups and identifying each \(U_J\) with its image in \(U_K\) whenever \(J\subseteq K\). This is the KMS group of type \(\widetilde A_n\) [23].

A finite quotient of the completion defines a KMS complex if its map is injective on every spherical local group and preserves their pairwise intersections; the complex is formed from the images of the maximal local groups by the common-intersection coset construction [23]. A diagram on \(n+1\) types is \(n\)-classical when every proper subdiagram has irreducible components of classical type \(A_r\), \(B_r\), \(C_r\) or \(D_r\) [23]. Thus \(\widetilde A_n\) is \(n\)-classical.

We use the following part of the theorem of Oppenheim–Valentiner-Branth [23]. For fixed \(n\ge3\) and sufficiently large \(q\), the \((n-1)\)-skeleta \(Y\) of finite \(n\)-dimensional, \(n\)-classical KMS complexes over \(\mathbb F_q\) satisfy, for some \(\eta>0\) independent of the finite quotient, \[ \norm{\delta_i f}_{Y,i+1} \ge\eta\,\operatorname{dist}_{Y,i}(f,Z^i(Y)) \qquad(0\le i<n-1). \tag{16}\] The weights are exactly (1). We only need this inequality, not the additional lower bound on nontrivial cosystoles.

The polynomial local groups above also appear in [23], where the finite quotients use irreducible polynomials. We verify the recognition conditions for our reductions modulo \(t^m\) directly. Our matrix model supplies a homomorphism from that completion to \(U\), and it is onto by 1. The local groups are realized faithfully by 1; their intersections are the groups belonging to the intersecting subsets.

The composite map to \(G_m\) has both properties by 1. Indeed the equality of intersections already holds in the universal completion: a member of two local groups has, in the matrix model, image in \(U_{J\cap K}\), and injectivity on either local group identifies it with the corresponding member of that common subgroup. Therefore \(T_m\) is an \(n\)-dimensional, \(n\)-classical KMS complex. This also explains why reduction modulo \(t^m\) is allowed: the definition requires a finite group quotient with the two stated local properties, and imposes no field condition on the quotient ring.

Choose \(q\) once and for all to satisfy \(q\ge5\), \(q\ge2^{n-1}\), and the largeness requirement for (16). There are arbitrarily large odd primes. Thus \[Y_m=\operatorname{skel}_{n-1}T_m\] satisfies (16) with a common constant \(\eta>0\). It remains to pass from these \((n-1)\)-skeleta to the \(d\)-skeleta \(X_m\) without losing a uniform expansion constant.

Degree bounds and growth

Recall \(M=q^{\binom N2}\) from (8). The common stabilizer of all base vertices is \(U_{\varnothing}=\{1\}\), so the chambers of \(T_m\) are in bijection with \(G_m\). A face of type \(A\) lies in exactly \(|U_{S\setminus A}|\) chambers. In particular, every vertex lies in exactly \(M\) chambers, and every panel (a face of dimension \(n-1\)) lies in exactly \(q\) chambers. Each chamber has \(N\) vertices, so every vertex of \(X_m\) lies in at most \[ D=M\binom{N-1}{d} \tag{17}\] faces of dimension \(d\). The complexes \(T_m\) and \(X_m\) are pure, and \(X_m\) is connected by 2.

The subgroup \(\{e_{12}(f):f\in R\}\) lies in \(U\) by definition, and its image in \(G_m\) has order \(q^m\). There are \(|G_m|/M\) vertices of each of the \(N\) types. Thus \[ |X_m(0)|=\frac{N|G_m|}{M}\ge\frac{Nq^m}{M}\longrightarrow\infty. \tag{18}\]

Weights on skeleta

The expansion theorem uses the intrinsic weights of \(Y_m\), while 1 uses those of \(X_m\). The following elementary observation keeps track of this change of normalization. Let \(T\) be a finite pure \(n\)-complex, put \(X=\operatorname{skel}_s T\), and write \(\mu_i=w_i^T\) and \(\nu_i=w_i^X\). For \(\tau\in T(s)\) let \(c(\tau)\) be its number of containing chambers. If \(0<a\le c(\tau)\le b\) for all \(\tau\in T(s)\), then, with \[\Lambda=\frac{\binom{n+1}{s+1}|T(n)|}{|T(s)|},\] we have \[ \frac{\Lambda}{b}\mu_i(\sigma)\le\nu_i(\sigma) \le\frac{\Lambda}{a}\mu_i(\sigma) \qquad(0\le i\le s). \tag{19}\] For \(i=s\), this follows from \(\nu_s(\tau)=\Lambda\mu_s(\tau)/c(\tau)\). For \(i<s\), use the averaging identities \[\mu_i(\sigma)=\binom{s+1}{i+1}^{-1} \sum_{\substack{\tau\in T(s)\\\sigma\subseteq\tau}}\mu_s(\tau), \qquad \nu_i(\sigma)=\binom{s+1}{i+1}^{-1} \sum_{\substack{\tau\in T(s)\\\sigma\subseteq\tau}}\nu_s(\tau).\] The first is a double count of chamber–face incidences, and the second is the definition of skeleton weights. Summing the bounds in degree \(s\) proves (19). The same inequalities hold for norms and distances to any common subspace.

Apply this first with \(T=T_m\) and \(s=n-1\). Every panel lies in \(q\) chambers, so \(a=b=\Lambda=q\) and the weights of \(Y_m\) are exactly the weights of \(T_m\) in their common degrees. Next take \(s=d\). Every \(d\)-face lies in between \(1\) and \(M\) chambers, so (19) gives \[ \frac{\Lambda_m}{M}\norm{f}_{Y_m,i} \le\norm{f}_{X_m,i} \le\Lambda_m\norm{f}_{Y_m,i}\qquad(0\le i\le d). \tag{20}\] Here \(\Lambda_m=\binom N{d+1}|T_m(n)|/|T_m(d)|\) is common to all degrees and cancels in the expansion inequality below.