Original Title: “The base change in the Atiyah and the Lück approximation conjectures”
Authors: Andrei Jaikin-Zapirain (ICMAT-UAM)
Source: Geometric and Functional Analysis 29 (2019), no. 2, 464538
Date of online publication: 2019
In 1976, Michael Atiyah introduced L2cohomology of manifolds with a free co-compact action of a countable group, like, for example, the universal cover of a compact manifold together with the action of the fundamental group of the manifold by deck transformations. He wanted to extend the Atiyah-Singer index theory of elliptic differential operators to the case of non-compact manifolds. Atiyah defined L2Betti numbers as von Neumann dimensions of the resulting L2-cohomology groups. Józef Dodziuk generalized the notion of L2Betti numbers to the context of groups acting on CW complexes. There is also an algebraic way to introduce L2-Betti numbers of a group G:
Let F be a free finitely generated group and let A ∈ Matn×m([F]) be a matrix over the group ring [F].
For each quotient G = F/N of F we can define a rank function rkG on matrices over [F] such that rkG(A) is the von Neumann rank of the l2-operator ΦG,A : l2(G)n → l2(G)m, obtained from the right multiplication by A. The numbers rkG(A)are called L2Betti numbers of G. For example, in the case where G is finite, The Lück approximation conjecture is a statement about convergence properties of L2Betti numbers. For example, one of its variations predicts that the map N → rkF/N (A) is continuous in the space of marked groups.
In the first result of this paper Andrei Jaikin-Zapirain proves the sofic Lück approximation conjecture. In particular, it is shown that the map N→ rkF/N (A) is continuous in the space of sofic marked groups. The notion of sofic groups was introduced by Mikhail Gromov. Informally, the class of sofic groups consists of groups whose Cayley graph can be approximated by finite graphs. No nonsofic group is known at this moment.
One immediate application of the sofic Lück approximation is that any sofic group satisfies the algebraic eigenvalue property. Observe that if G is finite and A is a square matrix over K[G](where K is a subfield of ), then any eigenvalue λ of ΦG,A is a root of the characteristic polynomial of ΦG,A, and so, it is algebraic over K. It is said that a group G satisfies the algebraic eigenvalue property if the eigenvalues of the operator ΦG,A are algebraic over K for any square matrix A with coefficients in K[G]. The algebraic eigenvalue conjecture, formulated by Jósef Dodziuk, Peter Linnell, Varghese Mathai, Thomas Schick, and Stuart Yates in 2003, claims that the algebraic eigenvalue property holds for an arbitrary group G.
In the second main result the author applies the sofic Lück approximation and shows that the strong Atiyah conjecture holds for a large class of groups including virtually compact special groups, Artin’s braid groups and torsionfree padic analytic prop groups.