The center-valued Atiyah conjecture for congruence subgroups of $\mathrm{SL}_n(\mathbb{Z})$ and their finite extensions

The center-valued Atiyah conjecture refines the strong Atiyah conjecture on $L^2$-Betti numbers: for a discrete group $G$ with $\mathrm{lcm}(G)<\infty$ it predicts that the center-valued $L^2$-dimensions $\dim^u(\ker A)$, $A\in\mathrm{Mat}_d(K[G])$, take values in the discrete additive subgroup $L_K(G)$ of the center of the group von Neumann algebra generated by the center-valued traces of projections over the finite subgroups of $G$. Knebusch, Linnell and Schick proved the conjecture for Linnell's class $\mathfrak{C}$ and for many groups approximable by such, but explicitly posed as open (their Question 1.12) the case of congruence subgroups of $\mathrm{SL}_n(\mathbb{Z})$ and their finite extensions, in particular the extensions containing torsion. The torsion-free case is the theorem of Farkas and Linnell and the $n=2$ case is virtually free, so the open content of the question is the torsion-containing case, for instance $\mathrm{SL}_n(\mathbb{Z})$ itself for $n\ge3$. The problem is to prove or refute the center-valued Atiyah conjecture for all finite extensions of congruence subgroups of $\mathrm{SL}_n(\mathbb{Z})$, equivalently to settle the Atiyah-expected Artinian structure of their division closures.

Background

For a discrete group $G$, the group von Neumann algebra $\mathcal{N}(G)$ is the closure of the complex group ring $\mathbb{C}[G]$ acting by left convolution on the Hilbert space $\ell^2(G)$. The canonical trace $\mathrm{tr}_G\colon\mathcal{N}(G)\to\mathbb{C}$, given by $\mathrm{tr}_G(a)=\langle a\delta_e,\delta_e\rangle$, extends to matrices and assigns to the kernel of any convolution operator $A\in\mathrm{Mat}_d(\mathbb{Z}[G])$ acting on $\ell^2(G)^d$ a von Neumann dimension $\dim_G(\ker A)$; these dimensions are the $L^2$-Betti numbers of free finite $G$-CW-complexes. The strong Atiyah conjecture emerged from the question of which real numbers occur as $L^2$-Betti numbers: when $G$ has finite subgroups of unbounded order even transcendental values can occur, so one restricts to groups whose finite subgroups have a finite least common multiple, denoted $\mathrm{lcm}(G)$. The conjecture over a subring $K\subseteq\mathbb{C}$ then predicts $\dim_G(\ker A)\in\frac{1}{\mathrm{lcm}(G)}\mathbb{Z}$ for every $A\in\mathrm{Mat}_d(K[G])$. It is known for large classes of groups, notably Linnell's class $\mathfrak{C}$, the smallest class of groups containing all free groups and closed under extensions with elementary amenable quotient and under directed unions of subgroups; here "elementary amenable" refers to the smallest class of groups containing all finite and all abelian groups and closed under subgroups, quotients, extensions and directed unions.

Knebusch, Linnell and Schick (KLS) introduced a center-valued refinement. Every finite von Neumann algebra carries a unique normalized center-valued trace $\mathrm{tr}^u\colon\mathcal{N}(G)\to\mathcal{Z}(\mathcal{N}(G))$, the center of $\mathcal{N}(G)$, through which every $\mathbb{C}$-valued trace factors; replacing $\mathrm{tr}_G$ by $\mathrm{tr}^u$ defines the center-valued dimension $\dim^u(V)=\mathrm{tr}^u(\mathrm{pr}_V)$. Because the center-valued dimension is a complete isomorphism invariant of finitely generated projective Hilbert $\mathcal{N}(G)$-modules, the refinement carries strictly more information than the scalar dimension. For $K\subseteq\mathbb{C}$ with field of fractions $F$ closed under complex conjugation, let $L_K(G)$ be the additive subgroup of $\mathcal{Z}(\mathcal{N}(G))$ generated by the elements $\mathrm{tr}^u(P)$, where $P$ runs over the projections in $F[H]$ and $H$ over the finite subgroups of $G$. The center-valued Atiyah conjecture asserts $\dim^u(\ker A)\in L_K(G)$ for all $A\in\mathrm{Mat}_d(K[G])$; it implies the strong Atiyah conjecture since $\mathrm{tr}_G=\mathrm{tr}_G\circ\mathrm{tr}^u$, and when $\mathrm{lcm}(G)<\infty$ the lattice $L_K(G)$ is discrete, so the conjecture predicts a quantization of the center-valued invariants. For torsion-free groups KLS observe that the center-valued conjecture is not stronger than the ordinary strong conjecture.

The conjecture has purely algebraic reformulations. Let $\mathcal{U}(G)$ denote the ring of (possibly unbounded) operators affiliated to $\mathcal{N}(G)$, and let $D(K[G])\subseteq\mathcal{U}(G)$ be the division closure of $K[G]$, the smallest subring of $\mathcal{U}(G)$ containing $K[G]$ and closed under taking inverses in $\mathcal{U}(G)$. Theorem 3.7 of KLS states that for $\mathrm{lcm}(G)<\infty$ and $K$ a subfield of $\mathbb{C}$ closed under complex conjugation the following are equivalent: $G$ satisfies the center-valued Atiyah conjecture over $K$; $D(K[G])$ is Atiyah-expected Artinian, meaning a semisimple Artinian ring (a finite direct product of matrix rings over skew fields) whose primitive central idempotents and matrix-summand sizes are explicitly dictated by the $K$-representation theory of the finite subgroups of $G$; and the canonical map $\bigoplus_{E\le G,\ |E|<\infty}K_0(K[E])\to K_0(D(K[G]))$ is surjective with $D(K[G])$ semisimple Artinian, where $K_0(R)$ is the Grothendieck group of finitely generated projective $R$-modules. For torsion-free $G$ this reduces to Linnell's 1993 theorem that the strong Atiyah conjecture is equivalent to $D(K[G])$ being a skew field.

KLS prove the center-valued conjecture for all groups in Linnell's class $\mathfrak{C}$ with $\mathrm{lcm}(G)<\infty$ and, for $K\subseteq\overline{\mathbb{Q}}$, for elementary amenable extensions of pure braid groups (the kernel of the natural map from the braid group $B_n$ onto the symmetric group $S_n$, that is, the fundamental group of the configuration space of $n$ ordered points in the plane), right-angled Artin groups (finitely generated groups whose only defining relations are commutation relations between some pairs of generators), primitive link groups (fundamental groups of tame links in the 3-sphere whose linking-number diagram is irreducible modulo some prime $p$, for instance when no linking number is divisible by $p$), and virtually cocompact special groups (fundamental groups of compact special cube complexes, a class containing hyperbolic groups acting cocompactly on CAT(0) cube complexes and fundamental groups of compact hyperbolic 3-manifolds with empty or toroidal boundary; the strong conjecture for these groups is due to Schreve). The proofs use either Linnell's extension technique or Knebusch's approximation theorem for center-valued $L^2$-Betti numbers together with the discreteness of $L_K(G)$.

For $N\ge3$ the principal congruence subgroup $\Gamma(N)=\ker(\mathrm{SL}_n(\mathbb{Z})\to\mathrm{SL}_n(\mathbb{Z}/N))$ is torsion-free, and every group containing $\Gamma(N)$ as a finite-index subgroup has $\mathrm{lcm}(G)<\infty$ (each of its finite subgroups injects into a suitable finite quotient). Farkas and Linnell proved the strong Atiyah conjecture for every torsion-free group containing such a congruence subgroup (in their setup, a $p$-adic congruence subgroup) as a finite-index normal subgroup, by embedding $\overline{\mathbb{Q}}[G]$ into a division ring inside $\mathcal{U}(G)$; by the preceding paragraph this also settles the center-valued conjecture in the torsion-free case. The case KLS explicitly leave open, their Question 1.12, is that of finite extensions containing torsion: for instance $\mathrm{SL}_n(\mathbb{Z})$ itself for $n\ge3$, which has property (T), a strong rigidity property of its unitary representations, and any subgroup strictly between $\Gamma(N)$ and $\mathrm{SL}_n(\mathbb{Z})$ containing a finite-order element. Such groups lie outside every class for which the conjecture is known, and the division-ring embedding argument of Farkas and Linnell uses torsion-freeness in an essential way. For $n=2$ the groups in question are virtually free, hence in $\mathfrak{C}$, so the open content of the question concerns $n\ge3$. The companion Conjecture 2.3 of KLS predicts that the division closure always coincides with the larger ring $\mathcal{E}(K[G])$ obtained by additionally adjoining central idempotents of $\mathcal{U}(G)$; KLS prove this equality under the center-valued conjecture, so a positive answer for a group $G$ would also settle that conjecture for $G$.

Problem Statement

Determine whether the following statement, posed as Question 1.12 of Knebusch, Linnell and Schick in its center-valued form, is true or false.

Let $n\ge2$ and $N\ge3$, let $\Gamma(N)=\ker(\mathrm{SL}_n(\mathbb{Z})\to\mathrm{SL}_n(\mathbb{Z}/N))$ be the principal congruence subgroup, and let $G$ be a discrete group containing a subgroup isomorphic to $\Gamma(N)$ as a subgroup of finite index; this is exactly the class of congruence subgroups of $\mathrm{SL}_n(\mathbb{Z})$ and their finite extensions, and $\mathrm{SL}_n(\mathbb{Z})$ itself belongs to it. Let $K\subseteq\mathbb{C}$ be a subfield closed under complex conjugation.

Prove, or refute, that $G$ satisfies the center-valued Atiyah conjecture over $K$: for every $d\in\mathbb{N}$ and every matrix $A\in\mathrm{Mat}_d(K[G])$ acting on $\ell^2(G)^d$ by left convolution,

$$\dim^u(\ker A)=\mathrm{tr}^u(\mathrm{pr}_{\ker A})\in L_K(G),$$

where $\mathrm{tr}^u$ is the center-valued trace on $\mathcal{N}(G)$ and $L_K(G)\subseteq\mathcal{Z}(\mathcal{N}(G))$ is the additive subgroup generated by the elements $\mathrm{tr}^u(P)$ with $P$ a projection in $F[H]$, where $H$ ranges over the finite subgroups of $G$ and $F$ is the field of fractions of $K$.

By Theorem 3.7 of KLS, for such $G$ the center-valued Atiyah conjecture over $K$ is equivalent to $D(K[G])$, the division closure of $K[G]$ in the ring of affiliated operators, being Atiyah-expected Artinian, and also to the canonical map $\bigoplus_{E\le G,\ |E|<\infty}K_0(K[E])\to K_0(D(K[G]))$ being surjective with $D(K[G])$ semisimple Artinian. The conjecture is well posed for the whole class since $\mathrm{lcm}(G)<\infty$ for every such $G$. The torsion-free subcase and the case $n=2$ are already known; the open core is the subcase in which $G$ contains torsion, for example $G=\mathrm{SL}_n(\mathbb{Z})$ with $n\ge3$, and the expected answer is affirmative.

References

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