A torsion-free counterexample to reduced Baum–Connes injectivity

1. Retain both constructions in a finitely generated subgroup

Embed the integral Heisenberg group $H=\langle x,y,z\mid[x,y]=z,[x,z]=[y,z]=1\rangle$ in $\Gamma_\nu$. Proposition 7.11 supplies torsion-freeness of $\Gamma_\nu$ and a class $c_{\Gamma_\nu}$ evaluating to $-1$ on the ordered torus $(z,x)$. Proposition 8.8 asserts integral vanishing of its Bott image for all sufficiently large $\nu$.

Lemma 8.9 approximates the finite matrix entries of a stabilization projection and a witnessing partial isometry by group polynomials. Their finite supports, together with generators of $H$, generate $G_{\mathrm{inj}}$. The reduced inclusion of a subgroup is isometric, so the small-defect bounds remain valid in this subgroup. Spectral cutting and polar normalization recover an exact witness there. Torsion-freeness and the embedded torus pass to the subgroup; setting $c=-c_{\Gamma_\nu}|_{BG_{\mathrm{inj}}}$ changes the evaluation to $1$. No limit group of the accuracy sequence is used.

Proposition 7.11 · p. 69 Proposition 8.8 · p. 80 Lemmas 8.7, 8.9 · pp. 80–81

2. Detect the topological class by an integer index

Choose a line bundle $L$ with $c_1(L)=c$. Naturality of the index pairing on the compact torus cycle gives

The tangent bundle is trivial, so the characteristic class in the index formula is $1$, and the untwisted Dirac index is zero. An integer-valued homomorphism taking $h$ to $1$ proves infinite order. Compactness of $BG_{\mathrm{inj}}$ is not required.

[BvE16, Theorem 1, p. 2] §9.2 · pp. 82–83

3. Identify the reduced assembly image of the same class

In the full-torus case of Emerson–Hudson, the dual subtorus is a point, giving $\mu_r^{\mathbb Z^2}([D_{T^2}])=\pm\beta$. The sign depends on orientation conventions and does not affect vanishing. Put $A=C_r^*(\mathbb Z^2)$, $B=C_r^*(G_{\mathrm{inj}})$ and let $j:A\to B$ be inclusion. The Miščenko bundle becomes $E_A\otimes_jB$. The manuscript checks on this torus cycle that extension of coefficients commutes with the index Kasparov product. The comparison theorem therefore gives

This concerns injectivity with trivial coefficients, a different assertion from known failures of surjectivity with nontrivial coefficients or the existence of a nontrivial projection in the scalar algebra.

[EH20, Theorem 1.3, p. 2] [KP20, Corollary 1.4; (1.6), pp. 5–7] §9.2 · pp. 82–83

1. Embed the cover and control finite-support chains

Unique recovery of long words and exclusion of short circuits let the manuscript prove, by plane diagrams, that a null-labeled source path has equal endpoints. The voltage cover $\widetilde X$ and its deck group $H$ consequently embed in the Cayley graph; translated sheets are indexed by $\Gamma/H$. The direct decomposition into sheet cycle coordinates makes the chain complex of the two-dimensional complex $Z$, obtained by coning each sheet, exact over any commutative coefficient ring.

The modules $C_1,C_2$ are free over $\mathbb Z\Gamma$. All chains and cycle decompositions have finite support; no independence of infinite sums is assumed.

Proposition 6.4 · p. 57 Lemmas 6.3, 7.3; Proposition 7.4 · pp. 55–57, 62–64 Lemma 7.6; (7.12) · p. 65

2. Rule out prime-order subgroups by resolution length

The group $H$ is torsion-free, cone-vertex stabilizers are conjugates of $H$, and other cell stabilizers are trivial. If $C_p\le\Gamma$, it acts freely on cells, and the chains over $\mathbb F_p$ give a length-two free resolution of the trivial $\mathbb F_p[C_p]$ module. But tensoring the periodic resolution over $\mathbb F_p[C_p]\cong\mathbb F_p[t]/(t^p)$ with the trivial module leaves homology $\mathbb F_p$ in every degree. Resolution comparison gives a contradiction. Hence there is no subgroup $C_p$ and $\Gamma$ is torsion-free.

Lemma 7.8 · p. 66

3. Extend an explicit cocycle to the ambient group

Use Heisenberg coordinates with the following product.

The integral 2-cocycle

evaluates to $-1$ on the bar cycle $[z|x]-[x|z]$. The module $K=\ker(C_0(Z)\to\mathbb Z)$ has a length-one free resolution, so $\operatorname{Ext}^2_{\mathbb Z\Gamma}(K,\mathbb Z)=0$. This makes $H^2(B\Gamma;\mathbb Z)\to H^2(BH;\mathbb Z)$ surjective. The manuscript also compares chain maps from the induced resolution and the bar resolution, identifying this map with actual subgroup restriction. An extension of $[\omega]$ therefore retains the required evaluation.

Lemma 7.9 · pp. 67–68 Lemma 7.10 · pp. 68–69

1. Realize finite Cayley formulas in full regular norm

Treat the localized differences $a_i=p_i-e$, adjacent-height rotations and rescalings as finite families of profiles. Proposition 5.2 imposes short-circuit exclusion and full regular operator-norm approximation on the same voltage choice. Small spectral-carrier traces and bounds on excluded coordinates control the loss when restricting to disjoint sheet domains. Products of incidences extract normalized constant modes, while a fixed gap $\rho<1$ suppresses their complements. The remaining Cayley edges form a forest, whose operator-norm error is $O(\sqrt{|S_{\mathrm{wt}}|}/q)$.

Lemma 8.4 obtains finite matrix group-algebra elements with arbitrarily small error. It does not assume physical height-selection projections or membership of Borel carriers in the reduced C* algebra. The order of choices is essential: first profiles, meshes and weight nets, then the degree parameter, and finally the initial height.

Proposition 5.2; §5.8 · pp. 41–42, 49–50 Lemmas 8.1–8.4 · pp. 72–76 §9.1 · pp. 81–82

2. Remove the initial term using two homotopies and finite additivity

Fix regular representations of $A_\nu=C_r^*(\Gamma_\nu)$ and put $Q_r=\prod_\nu A_\nu/\bigoplus_\nu A_\nu$, where products are bounded sequences and direct sums are norm-null sequences. Let $\mathcal B$ be the C* algebra generated by four bounded model families and two paths. Compression by almost-isometries gives an exact $*$-homomorphism $\Theta$ in the quotient. Uniform approximation on finite meshes proves $\Theta(\mathcal B)\subset Q_r$.

Denote the even, odd, positive-even and initial-height differences by $A_{\mathrm{ev}},A_{\mathrm{od}},A_{\mathrm{ev},+},A_0$. Using external unitization, set $b_I=[e+A_I]-[e]\in K_0(\mathcal B)$. Rotations on disjoint adjacent pairs, followed by rescaling, give

The second equality comes from a finite-size projection homotopy using orthogonal supports. The all-height families have already been constructed as bounded operators; the displayed argument does not sum infinitely many K-theory classes.

Lemma 8.5; (8.17)–(8.18) · pp. 76–77 (8.19)–(8.20) · pp. 77–78

3. Identify the physical Bott class and recover equality in one group

The initial profile and the subgroup Bott representative $a^{\mathrm{phys}}$ are related by a bounded $v$ satisfying $v^*va^{\mathrm{phys}}=a^{\mathrm{phys}}$ and $\Theta(A_0)=va^{\mathrm{phys}}v^*$. Supported transport in Lemma 8.6 constructs a fixed-size projection path from precisely these relations. Hence the physical Bott class vanishes in $K_0(Q_r)$.

Lemma 8.7 lifts a stabilization projection and a partial isometry from the quotient to bounded sequences, then uses spectral cutting and polar normalization at sufficiently late coordinates. This recovers equality in $K_0(A_\nu)$ with a fixed finite witness size. The unitary equivalence from right to left regular representations sends each group element to that same group element, yielding Proposition 8.8.

Lemma 8.6; (8.23)–(8.25) · pp. 79–80 Lemmas 8.7, 8.9 · pp. 80–81