A counterexample to the coarse Novikov conjecture

1 Introduction

Coarse index theory relates the large-scale geometry of a space to the K-theory of an algebra of controlled operators. Its assembly map sends geometric K-homology classes to their analytic indices. The injectivity question asks whether that index retains the information in a class after one has passed to large scales. Roe developed this viewpoint and its Novikov-type formulation [12]; Higson and Roe clarified the role of bounded geometry and coarsened K-homology [7]. Here we construct a bounded-geometry space for which ordinary reduced coarse assembly loses a class of infinite order.

A metric space X is uniformly discrete if distinct points have distance bounded below by a positive constant. It has bounded geometry if, for every R<\infty, the cardinalities of its radius-R balls have a common finite bound. For such a space, let P_r(X) be the Rips complex: its simplices are the finite subsets of X of diameter at most r. We use locally finite analytic K-homology and its coarsening,

Here and throughout, i is taken modulo two. The reduced, locally compact Roe algebra C^*(X) is the operator-norm closure of the locally compact finite-propagation operators on \ell^2(X)\otimes\HH_0, where \HH_0 is an infinite-dimensional separable Hilbert space. Section 2 specifies these conventions and the ordinary coarse assembly map

The coarse Baum–Connes conjecture asks whether this map is an isomorphism [7, Conjecture 6.6]. The rational coarse Novikov assertion is that \mu_X\otimes\id_{\Q} is injective. Passing through all Rips scales is essential: a cycle at one scale can bound at a larger scale and therefore represent zero in KX_*. An obstruction to injectivity must survive every such coarsening while its analytic index vanishes.

Yu proved coarse assembly isomorphism for bounded-geometry spaces of finite asymptotic dimension [17, Theorem 7.1], and subsequently for those admitting a coarse embedding into Hilbert space [18]. Skandalis, Tu and Yu placed the latter result in a groupoid framework [13]. Further positive injectivity results include coarse embeddings into uniformly convex Banach spaces [8, Theorem 1.1] and fibred coarse embeddings into Hilbert space [5, Theorem 34]. These theorems impose geometric hypotheses beyond bounded geometry. Our result shows that bounded geometry alone does not suffice.

We prove the following integral statement.

Predecessors and the scope of the obstruction

Yu already constructed a non-bounded-geometry example with a nontrivial coarse K-homology class of zero index, using spheres of increasing dimension and size [17, Section 8, Proposition 8.1]. Dranishnikov, Ferry and Weinberger constructed integral noninjectivity examples for a map from locally finite K-homology to Roe-algebra K-theory, also allowing spaces without bounded geometry [4, Introduction, pp. 920–921]. Their Remark 6.2 distinguishes those examples from the Rips-coarsened formulation: the relevant classes die under coarsening. Theorem 1.1 retains bounded geometry and detects its class in the actual direct limit.

Higson, Lafforgue and Skandalis constructed injectivity and surjectivity obstructions for several groupoid assembly maps [6]. Their Hausdorff groupoid example in Section 5 combines a degree-one Bott class with a Kazhdan projection to produce a kernel class. This is a close methodological antecedent of the winding and constant-vector averaging used here. Their coarse-space construction in Section 6 instead establishes non-surjectivity for specific congruence expander unions [6, Section 6, Proposition 11]. These conclusions must be separated: a groupoid kernel example does not itself provide the bounded-geometry space and ordinary Rips-domain kernel in Theorem 1.1.

Expander phenomena also coexist with injectivity. Willett and Yu proved ordinary coarse injectivity for coarse disjoint unions of finite graphs whose girth tends to infinity; with an additional weak-expansion hypothesis, surjectivity fails [14, Theorem 1.5]. The fibers in our construction have short cycles, so persistent boundary winding does not imply that large-girth hypothesis. More recent isomorphism results for certain relative expanders likewise depend on specific embedding hypotheses for the kernel and quotient families [3, Theorem 1.1]. The organization of our vertices into finite fibers does not assert such an embedding or a finite-group extension. Throughout, our target is the reduced, locally compact Roe algebra; maximal completions and coefficient-free group assembly are different questions. The independent companion article [11, Theorem 1.1] concerns group assembly, which for a torsion-free group maps the compactly supported K-homology of its classifying space to the K-theory of its reduced group algebra. It constructs a finitely generated torsion-free group for which this coefficient-free map is not rationally injective.

How the construction works

Begin with finite discs D_j of radii j in the triangular lattice, and let C_j be their boundary cycles. Above each vertex of D_j place a finite congruence quotient of \SL_3(\Z). The quotient is trivial on C_j and grows with depth inside the disc. Cayley edges connect vertices within a fiber; reduction maps and twisted matchings connect neighboring fibers. The resulting finite graph is denoted by X_j. Thus C_j is also a cycle in X_j.

Let B, D and X be the separated unions of the C_j, D_j and X_j, respectively. The family of oriented boundary classes gives an element of K_1^{\lf}(P_1(B)), and its image defines \alpha\in KX_1(X). Two features of the fibers have different roles.

First, the twisting prevents short loops in X_j from winding around the center once j is sufficiently large. At any prescribed Rips scale this produces an integral cocycle whose restriction to C_j has winding one. The associated K-theory pairing detects every nonzero integer multiple of \alpha at that scale. Sections 3 and 4 prove that \alpha has infinite order in the direct limit.

Second, property (T) uniformly approximates the projections onto constant vectors in all the finite fibers. Averaging into those vectors defines an isometry V:\ell^2(D)\otimes\HH_0\to\ell^2(X)\otimes\HH_0. The right translations used for twisting fix the constant vectors, so they do not interfere with this averaging. Section 5 proves that conjugation by V maps C^*(D) into C^*(X), despite the unbounded sizes of the fibers. On the boundary fibers this isometry is the ordinary inclusion. Since the boundary classes bound the filled discs, naturality of assembly then gives \mu_X(\alpha)=0 in Section 6.

The transfer argument isolates a useful principle: uniform finite-propagation approximation of constant-fiber projections, together with controlled operators between adjacent fibers, can produce a Roe-algebra homomorphism without a corresponding coarse map. This principle accounts for the difference between the persistent winding in X and the vanishing of its index through D.

Section 2 fixes the analytic K-homology, Rips metrics, and index naturality used by both branches. Section 3 constructs the common graph family and its short-loop estimate. Section 4 then proves persistence, independently of averaging. Section 5 proves the operator transfer, and Section 6 combines its exact boundary restriction with the filled-disc argument to annihilate the same class.

2 Coarse assembly and locally finite classes

We collect the functorial facts needed to carry a family of boundary classes through the construction. All spaces below are proper metric spaces, and all maps used to push forward locally finite K-homology are continuous and proper. A map between discrete metric spaces is coarse if it is proper and bornologous: for every R there is S such that points at distance at most R have images at distance at most S.

2.1 Rips complexes and their coarse structure

For a uniformly discrete bounded-geometry space Y, the complex P_r(Y) is locally finite and has uniformly bounded dimension at each fixed r. We use its usual simplicial topology. A concrete proper metric compatible with that topology and its coarse structure is convenient when the complex has disconnected components. Regard a point as a finitely supported probability measure on Y whose support has diameter at most r, and use the transportation metric

where \pi ranges over the couplings of \nu and \eta. On each finite subcomplex this metric gives the usual topology, equivalently the topology from spherical simplices. The inclusion y\mapsto\delta_y is isometric, and every point is within distance r of every vertex in its support. Moreover, a closed ball about \delta_y of radius R has all its supports in the finite set B_Y(y,R+r). Indeed, for x,z\in\supp(\nu), d_Y(y,x)\le d_Y(y,z)+r; averaging over z gives d_Y(y,x)\le d_r(\delta_y,\nu)+r. The closed ball is a closed subset of a finite subcomplex and is compact. Local finiteness gives the asserted topology globally.

In particular, choosing a support vertex gives a coarse inverse to the vertex inclusion. Distances between chosen support vertices and between the corresponding points differ by at most 2r. Thus (1) realizes the usual coarse structure determined by supports; it also supplies finite distances between disconnected simplicial components.

Comparison with spherical metrics. One can compare this metric directly with the standard semi-spherical extension: allow paths in spherical simplices, normalized to have edge length one, and jumps between vertices of cost d_Y. Write d_{\mathrm{ss}} for the infimum of their lengths. At fixed r, simplex dimensions have a common bound. On each simplex, the transportation distance is bounded by a constant depending only on r times its spherical distance; this follows by comparing barycentric coordinates in the bounded dimensions and using d_r\le (r/2)\norm{\nu-\eta}_1. The same inequality, with constant one, holds for vertex jumps. Conversely, one may travel to a support vertex, make one vertex jump, and travel to the endpoint. Thus, for a finite constant c_r,

The second inequality uses that the minimum distance between the two supports is at most d_r. The inequalities give the same coarse structure; the finite-subcomplex argument above gives the usual simplicial topology. These choices give the standard Rips coarsening and Roe-algebra identifications [15, Definition 7.2.8, Proposition 7.2.11 and Theorem 7.2.16]. The inclusions for increasing r are continuous proper coarse maps. A proper Lipschitz map of vertex spaces induces the corresponding proper simplicial maps of Rips complexes. The support-vertex selections are used for the Roe-algebra K-theory identifications; locally finite K-homology is pushed forward by the continuous proper simplicial maps. The selector compositions before and after an inclusion differ by a bounded distance, since their values lie in the same bounded support. This also applies to a proper Lipschitz map of vertex spaces, with the bound given by the diameter of its image support. Thus the Roe K-theory identifications respect these maps and the passage to larger scales.

2.2 Roe algebras and naturality

On an ample module over a proper metric space, an operator is locally compact if multiplication on either side by any function in C_0 makes it compact. Its propagation is at most R if its matrix coefficients between sets at distance greater than R vanish. For a discrete bounded-geometry space Y on \ell^2(Y)\otimes\HH_0, a finite-propagation operator is locally compact exactly when each of its matrix entries in \mathcal B(\HH_0) is compact. Indeed, the propagation bound leaves only finitely many nonzero entries in each row and column; finite-support functions give compact products, and C_0(Y) follows by uniform approximation.

The locally finite analytic index at each Rips scale takes values in the Roe algebra of that complex. Coarse equivalence with the vertex space identifies its K-theory with K_*(C^*(Y)). The compatible indices define \mu_Y. We use the usual reduced Roe index and its naturality [15, Sections 5.1 and 6.7, Lemma 7.1.3 and Theorem 7.2.16], with the corrected localization comparison in the authors’ errata [16, items 30–31]. All spaces used in this application are nonempty, and their ample modules may be chosen with infinite multiplicity. In particular, for a proper Lipschitz inclusion f:Y\to Z and its covering isometry F(\delta_y\otimes v)=\delta_{f(y)}\otimes v,

On vertex Roe algebras \Ad F is an actual homomorphism; the induced map on K-theory is independent of the covering choices. All the maps to which we apply (2) will be inclusions of this form.

2.3 Families and winding pairings

For a countable topological disjoint union of second-countable locally compact spaces Y_j, all the algebras C_0(Y_j) and their countable direct sum are separable. Countable additivity in the first variable of \KK [2, Theorem 19.7.1 and Section 19.8.10] gives

This is an integral product. Restriction to a clopen union of components is the corresponding coordinate projection, induced contravariantly by extension by zero on C_0.

For an oriented circle C, choose its analytic fundamental class [C] so that its index pairing with a unitary of winding one is 1. The circle index formula then reads

This fixes the otherwise harmless sign convention for the Dirac or Toeplitz description of the class [10, Example 4.2.4]. An integral simplicial 1-cocycle on a finite complex Y determines a class in H^1(Y;\Z), represented by a map f:Y\to S^1, and hence a unitary class [f]\in K_1(C(Y)). For an included oriented circle, naturality of the index pairing identifies its pairing with the sum of the cocycle around that circle [2, Proposition 18.7.1 and Corollary 18.10.3]. One can see the representing map directly: map vertices to 1, traverse each edge with the assigned integer winding, and extend over triangles because their winding sums vanish. Higher extensions exist since \pi_k(S^1)=0 for k\ge2.

3 Finite discs with congruence fibers

We construct finite graphs X_j containing distinguished boundary cycles C_j. The goal of this section is a uniform statement about short loops: at any fixed length bound, loops sufficiently far out in the family have zero signed crossing count for a fixed ray from the center of the underlying disc. This count agrees with winding whenever the projected loop avoids the center.

3.1 Congruence quotients and a separated holonomy

We need a quotient tower with uniformly bounded successive kernels, so that lifting base edges preserves bounded degree. We will also choose compatible twists whose trace separation rules out nonzero crossing count on short closed paths in sufficiently late graphs X_j. The choice of \SL_3(\Z) supplies both this quotient tower and the uniform averaging estimates used in Section 5.

Let \Gamma=\SL_3(\Z) and let S be the finite symmetric generating set of elementary matrices I\pm E_{ab}, a\ne b. To see that these generate, apply the Euclidean algorithm by determinant-one row operations to the primitive first column of an integral determinant-one matrix, then to its remaining 2-by-2 block. Each integral elementary operation is a power of a member of S. Put

with their reduction maps. All reductions from \Gamma, and between successive Q_k, are surjective. Indeed, over \Z/2^k\Z an invertible matrix has a unit in its first column. Elementary row operations move it to the first position and clear the remaining entries. Continuing reduces the matrix to a diagonal one. The identity

expresses the needed determinant-one row interchanges and diagonal matrices by elementary matrices. Every elementary matrix over the quotient lifts to an integral one, itself generated by S. There is a uniform bound

For k\ge1, a kernel element has the form I+2^kA modulo 2^{k+1} and is specified by the nine binary entries of A; the same bound holds for Q_1.

Choose a 2-adic unit t\in1+2\Z_2 transcendental over \Q. Such a choice exists because 1+2\Z_2 is uncountable and the algebraic elements over \Q are countable. For every nonzero integer m,

Otherwise multiplying the alleged equality by t^{\abs m} gives a nonzero polynomial over \Z satisfied by t. Define

These elements are compatible with reduction.

3.2 The base discs and the lifted edges

Use the triangular lattice with coordinates (a,b)\in\Z^2 and neighbor differences

Embed its two coordinate axes at angle \pi/3. Set

For j\ge2, let D_j be the finite induced graph on \rho\le j, equipped with its graph metric. Its elementary triangles fill a closed topological disc. The boundary C_j, given by \rho=j, is a cycle of length 6j; we equip its vertex set with its own cycle metric. When used as a topological circle, C_j denotes the geometric realization of this cycle. The depth of a vertex u\in D_j is

Depths of neighboring vertices differ by at most one.

Fix a ray from the origin meeting no other lattice vertex. For an oriented lattice edge (u,v) not incident to the origin, let a(u,v)\in\{-1,0,1\} be its signed intersection with this ray. Set a(u,v)=0 for edges incident to the origin. Choose the sign so that the positively oriented boundary has total charge one. Then a(v,u)=-a(u,v), and the sum of a on any closed edge path avoiding the origin equals its winding number.

The vertices of X_j are

Put undirected edges of length one as follows; repeated edges and loops may be discarded.

1. 1.

   Within the fiber over u, join q to sq for every s\in S.

2. 2.

   If (u,v) is an oriented base edge with equal depth k, join (u,q) to (v,qg_k^{a(u,v)}).

3. 3.

   If (u,v) goes from depth k+1 to depth k, join (u,q) to (v,\bar qg_k^{a(u,v)}), where \bar q denotes reduction. The reverse direction uses the inverse edge relation.

Left Cayley multiplication and right holonomy commute. This choice of sides will allow a short closed path to yield a conjugacy relation in a single quotient.

Each X_j is connected: the fibers are connected Cayley graphs, and every base edge has nonempty relations to both endpoint fibers. Each vertex has degree at most

Indeed, a base neighbor gives a bijection, a single parent, or at most h children. Boundary fibers are trivial, so the cycle C_j includes in X_j by u\mapsto(u,1). Figure 1 shows the depth labels and the two types of inter-fiber edge relation.

These graphs do have uniformly short cycles. In a fiber of depth k\ge1, the commuting generators A=I+E_{12} and B=I+E_{13} give the four distinct vertices q,Aq,BAq,Bq of a Cayley square. Thus the short-loop statement below concerns their charge, not the absence of short cycles.

3.3 Separated unions and their boundary maps

Choose b_j\in C_j. In each of the three unions

join the chosen basepoints of successive components by an auxiliary edge of length M_j=100+j^2; in X_j use (b_j,1). Use the induced path distance on the original vertex sets, without adding vertices along these auxiliary edges. Inside each component this is its original graph or cycle metric: leaving a component cannot provide a shortcut back into it. The distances between distinct components tend to infinity outside a finite family, so these are coarse disjoint unions.

The spaces are uniformly discrete and proper, and have bounded geometry. To check the last assertion directly, the weighted graph has degree at most \Delta+2 and every edge has length at least one. Hence a ball of radius R has at most \sum_{m=0}^{\lfloor R\rfloor}(\Delta+2)^m vertices. The same argument applies to B and D with their smaller degree bounds. There are proper 1-Lipschitz inclusions

Every path in B maps to a path of the same length, including the auxiliary links. Properness follows from injectivity and finiteness of bounded subsets in the target. The use of the intrinsic cycle metric on B ensures these Lipschitz assertions.

For later use, at every fixed distance bound R only finitely many pairs (u,v)\in D\times D from different components satisfy d_D(u,v)\le R. Such a path must cross a link of length at most R. There are only finitely many such links, and both endpoints of the pair lie within distance R of their endpoints. Those balls are finite.

3.4 Short loops have zero charge far out

For an edge path in X_j, define its charge to be the sum of a(u,v) on its projected base steps; fiber steps contribute zero. Charges are additive under concatenation and change sign under reversal.

The qualification “j sufficiently large” is part of the conclusion. It is precisely what is needed for a test at each fixed Rips scale; no single short-loop bound is asserted simultaneously for all components.

4 An infinite-order coarse K-homology class

The small-loop lemma converts winding on the base lattice into an integral cocycle on each sufficiently late Rips component. We use this cocycle to test the same boundary family at every scale.

All links between components of B have length greater than one, and each C_j has length 6j\ge12. Thus P_1(B) is exactly the topological disjoint union of the circles C_j. By (3), their positively oriented analytic classes determine

Let \beta\in KX_1(B) be its direct-limit image. For each integer r\ge1, let i_r:P_1(B)\to P_r(X) be the map induced by the boundary inclusion, and put

The classes \alpha_r represent one and the same element \alpha.

The component and the compactly supported test may change with r. The argument requires one test at each scale, and does not assert a single test persisting at all scales. We have proved that \alpha has infinite order integrally. Since the kernel of A\to A\otimes_{\Z}\Q is the torsion subgroup for any abelian group A, it follows that

No product in (3) has been rationalized separately.

5 Averaging defines a Roe-algebra homomorphism

We now prove the analytic input for vanishing. Averaging over each fiber will embed the module of the filled base discs into the module of X. The main point is to show that the resulting spatial homomorphism takes values in the locally compact Roe algebra, uniformly over growing fibers.

5.1 A general transfer criterion

The following criterion separates that operator argument from the particular congruence construction. Propagation of an operator between two fiber subspaces is measured using the ambient metric of X.

The amplified projection \widehat P in this proof is a bounded operator, not a locally compact Roe-algebra element. Its diagonal compression at x\in F_u is \abs{F_u}^{-1}1_{\HH_0}, which is not compact. The proof establishes local compactness for the sandwiches, where it is actually needed.

5.2 Uniform approximation from property (T)

Return to the finite fibers F_u=\{u\}\times Q_{k_j(u)} of our graph X. Kazhdan introduced property (T) and proved its higher-rank and lattice instances [9]. For \SL_3(\Z) it provides a constant \kappa>0 for the fixed generating set S: in any unitary representation without invariant vectors, each unit vector is moved by at least \kappa by some s\in S [1, Proposition 1.3.2 and Example 1.7.4].

Let \lambda_k denote the left regular representation on \ell^2(Q_k), and define

The sum retains multiplicities when elements of S have the same image in Q_k. Since S is symmetric, 0\le L_k\le2. Its kernel consists exactly of the constants, since S generates Q_k. On the orthogonal complement of the constants,

If p_k denotes projection onto the normalized constant vector, it follows that

For k=0 the approximation is exact. Take the direct sum of these lazy averaging powers over all fibers and call it P_n. They are contractions of propagation at most n, since each Cayley generator moves at most one edge in X. Equation (12) proves uniform norm convergence to the constant-fiber projection P. This verifies the first hypothesis of Proposition 5.1.

5.3 Controlled operators between fibers

It remains to verify the second hypothesis. Equal-depth edges act by the permutation prescribed by their right holonomy. For an edge from a shallow vertex u of depth k to a deep vertex v of depth k+1, put a=a(u,v) and h_k=\abs{\ker(Q_{k+1}\to Q_k)}. The graph relation gives an isometry J_a:\ell^2(Q_k)\to\ell^2(Q_{k+1}) and its adjoint:

Every coefficient lies on a graph edge, so these maps have propagation at most one. The child sets have the common cardinality h_k and are disjoint; this proves the isometry assertion. Since \abs{Q_{k+1}}=h_k\abs{Q_k}, both J_a and its adjoint carry the normalized constant vectors to the corresponding normalized constants: each deep coefficient in the image under J_a is (h_k\abs{Q_k})^{-1/2}, and each shallow coefficient in the image under J_a^* is h_k(h_k\abs{Q_{k+1}})^{-1/2}. These formulas also apply to the singleton quotient Q_0. The equal-depth permutations have the same property.

For u,v in the same base disc with d_D(u,v)\le R, choose a base path from v to u of length at most \lfloor R\rfloor. Composing the edge operators just described gives a contraction O_{uv} of propagation at most \lfloor R\rfloor, with O_{uv}\xi_v=\xi_u. This includes u=v by using the identity on its fiber.

For the cross-component pairs at distance at most R, use O_{uv}=\lvert\xi_u\rangle\langle\xi_v\rvert. There are only finitely many such pairs by Section 3, and their fibers are finite. Consequently their propagations have a common finite bound. Combining this bound with \lfloor R\rfloor supplies s_R in Proposition 5.1. We conclude that

is a *-homomorphism. This conclusion is an operator-norm statement in the reduced Roe algebra. It does not require V to cover a coarse map from D to X.

6 The boundary index vanishes

We finish by applying the transfer homomorphism to the boundary family constructed in Section 4. Only the actual boundary inclusions e:B\to D and i:B\to X use assembly naturality.

Let W_j be the filled triangulated disc underlying the lattice graph D_j, and put W=\coprod_{j\ge2}W_j. Every elementary triangle has vertices at pairwise distance one, so there is a factorization by continuous proper maps

The second map includes each elementary triangle as a Rips simplex. At scale one, no simplex of P_1(D) joins different D_j. A compact subset of this topological disjoint union therefore meets only finitely many of its finite components. Its inverse image in W is a closed subset of finitely many compact discs, proving properness of the second map; the first map is proper for the same reason. Each W_j is contractible; homotopy invariance in the first variable of \KK and \KK_1(\C,\C)=K_1(\C)=0 [2, Proposition 17.9.1 and Section 19.8.3] therefore give

Thus the whole family \beta_1 dies in K_1^{\lf}(P_1(D)), and

Write \HH_Y=\ell^2(Y)\otimes\HH_0 for Y=B,D,X. The covering isometries E:\HH_B\to\HH_D and I:\HH_B\to\HH_X of e and i are

Put e_{\mathrm{Roe}}=\Ad E and i_{\mathrm{Roe}}=\Ad I. Because the boundary fibers are the singleton Q_0, the averaging isometry of Section 5 satisfies the exact identity

It follows on the entire Roe algebra C^*(B) that

The following diagram displays the naturality square for e and its operator-algebra continuation. The lower composite equals (i_{\mathrm{Roe}})_* by (16):

Naturality for i, followed by (16), naturality for e, and (15), now gives

The class here is exactly \alpha=i_*\beta tested at every Rips scale in Proposition 4.1. That proposition makes it an infinite-order class, while (9) makes \alpha\otimes1 nonzero in the rationalized domain. Its rational index is zero as well. This proves Theorem 1.1.