The strong Atiyah conjecture for right-angled Artin and Coxeter groups. (Q431056)

The strong Atiyah conjecture is about the possible values of the \(L^2\)-Betti numbers of a finite CW-complex. The following is an equivalent formulation for a CW-complex with fundamental group \(G\). Assume that \(G\) has a bound on the orders of its finite subgroups, let \(\text{lcm}(G)\) be the least common multiple of the orders of the finite subgroups of \(G\) and \(K\subset\mathbb C\) be a subring. The group \(G\) satisfies the strong Atiyah conjecture over \(K\) (or \(K[G]\) satisfies the strong Atiyah conjecture) if for every matrix \(A\in M_n(K[G])\) it follows that \[ \dim_G(\ker(A)):=\text{tr}_G(\text{pr}_{\ker(A)})\in\frac{1}{\text{lcm}(G)}\mathbb Z, \] where \(\text{pr}_{\ker(A)}\) denotes the orthogonal projection onto the kernel of \(A\). The main theorem is the following. Theorem. Let \(H\) be a right-angled Artin group or the commutator subgroup of a right-angled Coxeter group, let \(1\to H\to G\to Q\to 1\) be an extension with \(Q\) elementary amenable and such that \(\text{lcm}(G)<\infty\). In case \(H\) is the commutator subgroup of a right-angled Coxeter group, assume that all finite subgroups of \(Q\) are \(2\)-groups. Then \(G\) satisfies the strong Atiyah conjecture over \(\overline{\mathbb Q}\). This theorem includes both right-angled Artin and right-angled Coxeter groups.