The base change in the Atiyah and the Lück approximation conjectures

Abstract

Let F be a free finitely generated group and $${A \in {\rm Mat}_{n \times m}(\mathbb{C}[F])}$$ . For each quotient G = F/N of F we can define a von Neumann rank function rkG(A) associated with the l2-operator l2(G)n → l2(G)m induced by right multiplication by A. For example, in the case where G is finite, $${{\rm rk}_G(A)=\frac{{\rm rk}_{\mathbb{C}}({{\bar{A}}})}{|G|}}$$ is the normalized rank of the matrix $${{{\bar{A}}} \in {\rm Mat}_{n \times m}(\mathbb{C}[G])}$$ obtained by reducing the coefficients of A modulo N. One of the variations of the Luck approximation conjecture claims that the function $${N\mapsto {\rm rk}_{F/N}(A)}$$ is continuous in the space of marked groups. The strong Atiyah conjecture predicts that if the least common multiple lcm(G) of the orders of finite subgroups of G is finite, then $${{\rm rk}_G(A) \in \frac{1}{lcm (G)}\mathbb{Z}}$$ . In our first result we prove the sofic Luck approximation conjecture. In particular, we show that the function $${N \mapsto {\rm rk}_{F/N}(A)}$$ is continuous in the space of sofic marked groups. Among other consequences we obtain that a strong version of the algebraic eigenvalue conjecture, the center conjecture and the independence conjecture hold for sofic groups. In our second result we apply the sofic Luck approximation and we show that the strong Atiyah conjecture holds for groups from a class $${{\mathcal{D}}}$$ , virtually compact special groups, Artin’s braid groups and torsion-free p-adic analytic pro-p groups.