An irrational-trace counterexample to reduced Baum–Connes

1. From simultaneous overlap control to the group and its local distributions

Choose expanders $\Theta_i$ of fixed degree $d$, with a common Laplacian gap $\rho>0$ and growing girth $\sigma_i$. Put $n_i=|\Theta_i|$ and $k_i=\lfloor n_i/2\rfloor$, passing to a subsequence so that $k_i$ strictly increases and $k_{i+1}-k_i\to\infty$. The LPS family supplies the bounded degree, growing girth and bounded diameter/girth ratio required by Osajda's labelling theorem. Its finite alphabet $S$ defines a graphical presentation of a group $\Gamma$ in whose Cayley graph $X$ the $\Theta_i$ embed isometrically.

[LPS88, Theorems 3.4 and 4.1, print pp. 267, 269 / PDF pp. 7, 9] [Osa20, Theorem 2.7, print p. 173 / PDF p. 15] [Osa20, Lemma 3.1; Theorem 3.2, print p. 176 / PDF p. 18] Proposition 3.1 · pp. 9–10

Cone off every translated copy $P=g\Theta_i$ by adding a center $c_P$. The manuscript bounds the number of arcs in compressed diagrams, obtains a uniform coarse linear filling inequality, and invokes Bridson–Haefliger to obtain a hyperbolicity constant independent of the labels and $\lambda$. Only then is $\lambda$ chosen small. For a root $R$ and a copy with $c_P\ne R$, the intersections of $P$ with all other copies whose centers are no farther from $R$ (including equal distances) consequently lie in one intrinsic induced tree ball $B_P$. Its radius is at most $\sigma(P)/16$ and it has fewer than $\eta|P|$ vertices, where $\eta=1/100$. Pairwise small intersections alone do not supply this simultaneous containment.

[BH99, III.H.2.1; III.H.2.9, print pp. 414, 419 / PDF pp. 437, 442] Proposition 2.3; Lemma 2.4 · pp. 7–9

Fix one sufficiently large prime $p>2$ and encode each allowed set of $d$ outgoing labels by a nonzero element of $\mathbb F_p$. Write $w_P$ for the prescribed word on a copy. Lemma 4.1 constructs a code $w_P\in C_P\subset\mathbb F_p^P$ of dimension $k_i$, with every nonzero word in either $C_P$ or $C_P^\perp$ supported on more than $\eta|P|$ vertices. Existence follows by choosing a random $k_i$-dimensional subspace containing $w_P$: the number of low-weight words is less than $p^{n_i/4}$, and a union bound makes the probability of a low-weight primal or dual word less than one. Impose all translated dual relations:

The subspace $W_0$ is $\Gamma$-invariant, and $V$ is generated by the $\Gamma$-orbit of $v_*=[e_1]$. Thus the lifts of $S$, together with $(v_*,1)$, generate $G_{\mathrm{sur}}$.

Lemma 4.1 · pp. 12–13 Lemma 4.2; Definition 4.3 · pp. 13–14

The key equality is $W_0\cap\mathbb F_p^P=C_P^\perp$. If a finite sum supported on $P$ used a relation from another copy, choose a participating copy $Q$ farthest from the root $c_P$. All its intersections with the other supports lie in $B_Q$, whereas the dual word has larger support. A coordinate outside this ball cannot cancel, a contradiction. Dualizing to $\Omega=\operatorname{Hom}_{\mathbb F_p}(V,\mathbb F_p)$ shows that restriction $\omega\mapsto X_\omega|_P$ has image exactly $C_P$, and pushes Haar measure to its uniform distribution. The prescribed pattern therefore has probability $p^{-k_i}$.

Lemma 4.2; Definition 4.3 · pp. 13–14 Lemma 4.4 · pp. 14–15

The uniform gap for the operator $T$ in the next section gives $b_0=1_{\{0\}}(T)\in C_r^*(G_{\mathrm{sur}})$. This is an actual projection in the ordinary reduced algebra, not merely a class in a matrix algebra or a projection available only in the von Neumann algebra.

Proposition 5.5 · p. 21

2. Canceling copy counts against diagonal entries

Call a copy valid when it has the prescribed pattern. Valid copies are disjoint isolated components of the enabled graph, and $\pi_\omega(b_0)$ projects onto constants on each of them and vanishes on the remainder. Its diagonal entry on a type $i$ copy is $1/n_i$. Exactly $n_i$ type $i$ copies contain the identity: their translations are $g=\varphi_i(v)^{-1}$ for the embedded vertices $v$. The events that these copies are valid are mutually exclusive, including across types. Hence

No independence between copies is needed. Since $p>2$, the base-$p$ expansion has digit one exactly at the positions $k_i$. The infinitely many ones have diverging gaps, so this expansion is not eventually periodic. The trace is therefore irrational and lies strictly between zero and one.

Section 5.1, (17)–(21) · pp. 15–16 Proposition 6.1 · pp. 21–22

3. Excluding the class from proper-action assembly

For any countable discrete group $G$, set $\Lambda_G=\mathbb Z[1/|F|:F\leq G\text{ finite}]\subset\mathbb Q$. Lück's Theorem 0.3 identifies the image of localized assembly followed by the canonical trace with $\Lambda_G$. Evaluating at $1\otimes x$ for an integral class $x$ yields

Neither surjectivity of assembly nor injectivity of localization is required. This is the manuscript's Theorem 1.2. Lück's Section 4 uses the direct limit of equivariant KK-groups over cocompact subcomplexes of a proper space, and Section 5 uses its analytic index. The manuscript employs this proper-action assembly.

[LU02, Theorem 0.3, p. 2] [LU02, Sections 4–5, pp. 23, 25] Theorem 1.2 · p. 2

The irrational trace of $[b_0]$ excludes it from the image. The identification $C_r^*(V\rtimes\Gamma)\cong C(\Omega)\rtimes_r\Gamma$ is used to analyze the operator; the obstructed map is the coefficient-free assembly map for $G_{\mathrm{sur}}$ itself. A finite rational linear combination of assembly classes still has rational trace, so $[b_0]\otimes1$ also lies outside the rationalized assembly image, as observed in the unnumbered consequence on manuscript p.22.

Proof of Theorem 1.1 and rationalization · p. 22

1. Counting local deficits without overlap

On an invalid copy, the support of $X_\omega|_P-w_P\in C_P\setminus\{0\}$ exceeds $\eta|P|$. Every discrepancy removes at least one prescribed edge incidence. Combining these missing incidences with the expander boundary estimate gives $d|B|-2e_{\omega,P}(B)\geq c|B|$ for every $B\subset P$, where $c=(\eta/2)\min\{\kappa,1\}>0$ and $\kappa$ is the edge-expansion constant. Overlaps prevent simply summing this inequality over copies.

Lemmas 5.2–5.3 · pp. 16–19

The finite-support cycle space decomposes as $Z_1(X;\mathbb F_2)=\bigoplus_P Z_1(P;\mathbb F_2)$. For the enabled graph $A$ on a finite set $Y\subset U_\omega$, eliminate its cycle coordinates in decreasing order of distance from a root. Coordinate projections are not restrictions of edge chains, but any extra edge arising through cancellation with remaining copies lies in $B_P$. The projected coordinate therefore belongs to the cycle space of $D_P=B_P\cup(A\cap P)$.

Lemma 3.2 · p. 11 Lemmas 5.2–5.3 · pp. 16–19

If a component $C$ outside $B_P$ has $l_C$ attachment edges to the ball, its contribution is $b_1(C)+\max(0,l_C-1)$. The girth and radius bounds separate distinct attachment endpoints by more than $\sigma(P)/2$, bounding the extra rank by $4|C|/\sigma(P)$. The local deficit and $\sigma(P)>20/c$ give

The exterior sets $(Y\cap P)\setminus B_P$ are pairwise disjoint in the elimination order. Their charges can therefore be summed. Euler's formula then gives the global deficit

This charging to exterior vertices is the central step from individual expander estimates to the whole graph.

Lemmas 5.2–5.3 · pp. 16–19

2. A uniform lower bound on the remaining components

On $U_\omega$, retain the vertex potential $q(x)=d-\deg_\omega(x)\geq0$. Integrating the finite-set deficit over level sets of $|\xi|^2$ and applying Cauchy–Schwarz jointly to the edge and potential terms gives

Each valid copy has gap $\rho$ off its constants, and no enabled edge joins different valid copies or the remainder. Thus, uniformly in $\omega$,

Lemma 5.4 · pp. 20–21

3. Returning through a faithful representation to continuous functional calculus

The direct sum $\bigoplus_\omega\pi_\omega$ of the regular representations induced from evaluations is faithful on the reduced crossed product. Uniformity of the gap prevents nonzero spectrum from approaching zero in this direct sum. Consequently $T$ has the same gap in $C_r^*(G_{\mathrm{sur}})$. Choose a continuous function with $\chi(0)=1$ and $\chi|_{[\varepsilon,2d]}=0$. Then $b_0=\chi(T)$ is a projection in that C*-algebra. Functional calculus commutes with every $\pi_\omega$, yielding the asserted kernel and diagonal descriptions.

Section 5.1, (17)–(21) · pp. 15–16 Proposition 5.5 · p. 21