A torsion-free counterexample to the Kadison–Kaplansky projection conjecture

1. Establishing the group hypotheses separately from the analytic output

The voltage group $F$ is a nilpotent torsion-free group with integer coordinates. It is a central product of two integer affine groups, identifying their centers with opposite signs. The identified subgroup is primitive in the central $\mathbb Z^2$, leaving an integer central coordinate. Positive powers multiply the first nonzero graded coordinate by that positive integer, excluding torsion. This alone does not imply that the presented group $\Gamma_{\mathrm{proj}}$ is torsion-free. Theorem 7.1 uses long-word recovery and a girth condition on the cover to obtain embedded copies, a single finite ball containing each copy's earlier intersections, and a forest of remaining edges. Proposition 7.14 uses the embedding of copy stabilizers in $F$ and the acyclicity of the cone complex. A subgroup of order $p$ would give the trivial module over $\mathbb F_p[C_p]$ a free resolution of length two, contradicting the periodic resolution with nonzero homology in arbitrarily high degrees. Finite generation follows from finiteness of the final alphabet.

Lemma 8.3; equation (8.5) · pp. 40–42 Theorem 7.1 · pp. 26–27 Proposition 7.14 · pp. 36–38

2. Obtaining an actual projection before passing to the free product

Theorem 10.11 supplies an actual projection $p_0\in M_N(C_r^*(\Gamma_{\mathrm{proj}}))$, not just an abstract $K_0$ class. The matrix trace is

without division by $N$. Its value lies in $(0,1/2)$, so Proposition 2.2 applies. The free product $G_{\mathrm{proj}}=\Gamma_{\mathrm{proj}}*F_N$ preserves finite generation and torsion-freeness, and its constructed scalar projection has the same trace.

Theorem 10.11 · p. 72 Section 11 · p. 73

1. Combining expansion with trace decay at both endpoints

The tower $Y^0$ before pruning in Proposition 6.3 preserves normalized constants on each bipartite incidence and contracts their orthogonal complements by at least a uniform gap $\chi$. This constant-preservation property is not assumed unchanged for the pruned graph $Y$; the subsequent comparison uses the compression $\Pi O\Pi$. The vertex fraction $\beta_\ell$ lost in the pruning needed for label recovery tends uniformly to zero. To select expanding finite blocks, the proof estimates the dependency neighborhoods and probabilities of bad-cut events and applies the asymmetric Lovász local lemma. It does not choose a different final parameter for every layer of the infinite tower.

Proposition 6.3 · pp. 24–25 Proposition 5.4 · pp. 18–20 [MT10, Theorem 1.1, p. 1]

On the analytic side, for the central Fourier variable $t$ of $F$, the construction combines a twisted-convolution projection $p_t$ with $p_{t-\delta}$ shifted by a finite magnetic representation. Here $n_f=m(m-1)/2$ counts the strictly upper triangular coordinates used in the construction. With $\delta=q^{-1}$ and $N=q^{n_f}$, for $0<t<\delta$ one has

Absorbing the finite magnetic representation contributes the factor $N$ to the trace. Decay at both endpoints must survive this factor. Windows satisfying $\sum_i h_i(t)^2=1$ occupy at most two adjacent layers; their supporting layers escape toward the two ends of the tower as $t$ approaches the endpoints. A symmetric combination of normalized constant vectors gives the model $O(t)$ the eigenvalue $+1$. Expansion bounds all remaining positive spectrum by $1-\chi$.

Proposition 8.7 · pp. 45–46 Lemma 8.9; equations (8.20)–(8.21) · pp. 47–48 Lemma 9.1 · pp. 50–51

2. Enforcing girth and regular-norm approximation with the same voltages

Theorem 9.2 simultaneously chooses edge voltages $z_e\in F$ and finitely many matrix weights $W_e$ so that the cover has the required girth and

Here $\Pi$ retains the surviving vertices and $\mathcal S$ is the finite alphabet augmented by weight tags. Lemma 9.3 averages compressions to coordinate boxes in the nilpotent group, transferring finite matrix tests back to the full regular operator norm. Centered-moment estimates and bounds on the probability that a short closed walk has trivial holonomy make the total failure probability over all layers less than one. The claimed gap is not inferred merely from selected finite-dimensional representations.

Theorem 9.2 · pp. 52–63

3. Preserving the spectral gap with a nonequivariant domain assignment

Compression and a small perturbation of the model produce the cover's upper projection $E_P$. Let $e_v$ be its identity diagonal summed over matrix indices at a base vertex, and let $w_i=2^{C_\bullet H_i}$ be the geometric ball-size weight. Endpoint decay and decay of the resolvent across layers yield

There is no extra layer-cardinality or matrix-dimension factor in this estimate. Removing earlier intersections from a copy $Q$ produces disjoint domains $D_Q$ and loses at most $\sqrt S$ in the norm of its upper projection. The remaining edges form a forest, so their weighted adjacency has norm at most $4C_\eta\sqrt{|\mathcal S|/(dD)}=o(1)$. These two small errors give, with $\alpha=\chi/1000$,

The upper part is nonempty, and $1-\chi/4$ lies in the gap. Continuous functional calculus therefore places $p_0=1_{[1-\chi/4,\infty)}(T)$ inside the reduced C-algebra*. Existence of a spectral projection in the von Neumann algebra alone would not establish this membership.

Proposition 8.11 · pp. 49–50 Propositions 10.3–10.4 · pp. 65–67 Proposition 10.7 · pp. 68–69

4. Bounding the trace through an equivariant map to all copies

The assignment of $D_Q$ need not be equivariant, so the preceding ordinary partial-isometry comparison does not by itself compare canonical traces. Set $\mathcal H=\ell^2(\Gamma_{\mathrm{proj}})\otimes\mathbb C^N$. Instead use all indexed copies and define $\mathcal A\xi=(E_Q(\xi|_Q))_Q$. The equivariant incidence bijection $(Q,x)\leftrightarrow(x,v)\in\Gamma_{\mathrm{proj}}\times V(Y)$ identifies the target upper space's identity trace with $S_0=\sum_v e_v\le S$.

The overlap-loss bound makes $\mathcal A$ bounded. On $K=\ker\mathcal A$, one has $\langle T\xi,\xi\rangle\le(1-\chi+3\alpha)\|\xi\|^2$, whereas the lower bound on the range of $p_0$ is $1-3\alpha$. Hence $P_{K^\perp}:p_0\mathcal H\to K^\perp$ is injective. Equivariant polar decomposition gives

Strict positivity follows from $p_0\ne0$ and faithfulness of the trace. Geometry, magnetic representation, and approximation accuracy are fixed in order, followed by one sufficiently large finite $\ell$. The final step takes neither a limit group nor a limit projection.

Lemmas 10.8–10.10 · pp. 70–72 Theorem 10.11 · p. 72

1. Controlling cancelling inputs by the trace

Subgroup inclusion preserves the reduced norm and trace. In coordinate $j$ of $\xi=p\xi$, call an input bad if its normal form starts with $t_j^{-1}$ and good otherwise. Good outputs have distinct initial letters $t_j$, hence disjoint supports. Each bad input position belongs to a unique bad coordinate. Cauchy–Schwarz using the projection's diagonal coefficients gives

Thus $\|U\xi\|\ge c_a\|\xi\|$ with $c_a=\sqrt{1-a}-\sqrt a>0$. A normalized matrix trace cannot replace $a$ in this estimate.

Lemma 2.1 · p. 4 equations (2.3)–(2.7) · p. 5

2. Constructing the actual projection using the corner inverse

The inequalities $c_a^2p\le A\le Np$ imply that $A^{-1}$ and $A^{-1/2}$ belong to the same C*-corner. The row $V=UpA^{-1/2}\in M_{1,N}(C_r^*(K))$ satisfies $V^*V=p$, and $e=VV^*$ is a projection in the scalar algebra. Traciality gives

The inequalities $0<a<1/2$ exclude both $e=0$ and $e=1$. This construction uses functional calculus in the corner and a finite row; it does not infer the existence of a projection merely from comparison of $K_0$ classes.

equation (2.2) and proof · pp. 5–6

3. Preserving the group conditions and completing the theorem

Take $\Lambda=\Gamma_{\mathrm{proj}}$ and $p=p_0$. Adjoining finitely many free generators preserves finite generation. Every nontrivial free-product element is conjugate either into a factor or to a cyclically reduced word of length at least two; positive powers of the latter have increasing length. As both factors are torsion-free, the free product is torsion-free. This supplies the group conditions and all projection conclusions of Theorem 1.1.

Proposition 2.2 · pp. 4–6 Section 11 · p. 73