Approximating L2-invariants, and the Atiyah conjecture

Abstract

Let G be a torsion free discrete group and let \bar{Q} denote the field of algebraic numbers in C. We prove that \bar{Q}[G] fulfills the Atiyah conjecture if G lies in a certain class of groups D, which contains in particular all groups which are residually torsion free elementary amenable or which are residually free. This result implies that there are no non-trivial zero-divisors in C[G]. The statement relies on new approximation results for L2-Betti numbers over \bar{Q}[G], which are the core of the work done in this paper. Another set of results in the paper is concerned with certain number theoretic properties of eigenvalues for the combinatorial Laplacian on L2-cochains on any normal covering space of a finite CW complex. We establish the absence of eigenvalues that are transcendental numbers, whenever the covering transformation group is either amenable or in the Linnell class \mathcal{C}. We also establish the absence of eigenvalues that are Liouville transcendental numbers whenever the covering transformation group is either residually finite or more generally in a certain large bootstrap class \mathcal{G}. Please take the errata to Schick: "L2-determinant class and approximation of L2-Betti numbers" into account, which are added at the end of the file, rectifying some unproved statements about "amenable extension". As a consequence, throughout, amenable extensions should be extensions with normal subgroups.

AMS-LaTeX2e, 33 pages; improved presentation, new and detailed proof about absence of trancendental eigenvalues; v3: added errata to "L2-determinant class and approximation of L2-Betti numbers", requires to restrict to slightly weaker statements

References in corpus (1)

• Approximating L2-invariants, and the Atiyah conjecture

Cited by in corpus (48)

• Approximating L2-invariants, and the Atiyah conjecture
• Spectral Analysis of Percolation Hamiltonians
• Noncommutative knot theory
• Rational group ring elements with kernels having irrational dimension
• Hamiltonians on discrete structures: Jumps of the integrated density of states and uniform convergence
• On Turing dynamical systems and the Atiyah problem
• Hodge Theory on Metric Spaces
• The strong Atiyah conjecture for right-angled Artin and Coxeter groups
• Finite group extensions and the Atiyah conjecture
• Integrated density of states and Wegner estimates for random Schrödinger Operators
• Closed manifolds with transcendental L2-Betti numbers
• The strong Atiyah and Lück approximation conjectures for one-relator groups
• A Banach space-valued ergodic theorem and the uniform approximation of the integrated density of states
• The Atiyah conjecture and Artinian rings
• Spectral and Combinatorial Aspects of Cayley-Crystals
• Uniform existence of the integrated density of states for random Schrödinger operators on metric graphs over ${\mathbb{Z}}^d$
• Continuity properties of the integrated density of states on manifolds
• The free field: realization via unbounded operators and Atiyah property
• Equality of Lifshitz and van Hove exponents on amenable Cayley graphs
• A linear Wegner estimate for alloy type Schroedinger operators on metric graphs
• Three flavors of twisted invariants of knots
• Uniform existence of the integrated density of states for combinatorial and metric graphs over Z^d
• L^2-spectral invariants and convergent sequences of finite graphs
• $L^2$-spectral invariants and quasi-crystal graphs
• Group cocycles and the ring of affiliated operators
• Non-commutative Multivariable Reidemeister Torsion and the Thurston Norm
• Convergence theorems for graph sequences
• Thurston norm via Fox calculus
• Applications of Foelner's condition to quantum groups
• An uncertainty principle and lower bounds for the Dirichlet Laplacian on graphs
• The Kazdan-Warner equation on canonically compactifiable graphs
• Reidemeister torsion, the Thurston norm and Harvey's invariants
• Approximation of center-valued Betti-numbers
• Traces on group C*-algebras, sofic groups and Lück's conjecture
• On computing homology gradients over finite fields
• Uniform existence of the integrated density of states for randomly weighted Hamiltonians on long-range percolation graphs
• The Grothendieck group of polytopes and norms
• L2-torsion, the measure-theoretic determinant conjecture, and uniform measure equivalence
• Approximation of the integrated density of states on sofic groups
• The Strong Approximation Conjecture holds for amenable groups
• Groupoids, von Neumann Algebras and the Integrated Density of States
• Uniform existence of the integrated density of states on metric Cayley graphs
• The parity of the Cochran-Harvey invariants of 3-manifolds
• Integer operators in finite von Neumann algebras
• Equivariant Benjamini-Schramm Convergence of Simplicial Complexes and ${\ell }^2$-Multiplicities
• L^2-Betti numbers of plane algebraic curves
• Elliptic operators on planar graphs: Unique continuation for eigenfunctions and nonpositive curvature
• Arithmetic properties of eigenvalues of generalized Harper operators on graphs