Integrality of L2-Betti numbers

Abstract

The Atiyah conjecture predicts that the L2-Betti numbers of a finite CW-complex with torsion-free fundamental group are integers. We show that the Atiyah conjecture holds (with an additional technical condition) for direct and inverse limits of directed systems of groups for which it is true. As a corollary it holds for residually torsion-free solvable groups, e.g. for pure braid groups or for positive 1-relator groups with torsion free abelianization. Putting everything together we establish a new class of groups for which the Atiyah conjecture holds, which contains all free groups and in particular is closed under taking subgroups, direct sums, free products, extensions with elementary amenable quotient, and under direct and inverse limits of directed systems. This is a corrected version of an older paper with the same title. The proof of Proposition 2.1 of the earlier version contains a gap, as was pointed out to me by Pere Ara. This gap could not be fixed. Consequently, in this new version everything based on this result had to be removed. Please take the errata to "L2-determinant class and approximation of L2-Betti numbers" into account, which are added, rectifying some unproved statements about "amenable extension". As a consequence, throughout, amenable extensions should be extensions with normal subgroups.

25 pages, one unclear sentence in Abstract changed. v3: errata to "L2-determinant class and approximation of L2-Betti numbers" added which require to replace some of the results by apparently slightly weaker versions

References in corpus (4)

• Approximating L2-invariants, and the Atiyah conjecture
• On a conjecture of Atiyah
• L2-determinant class and approximation of L2-Betti numbers
• Integrality of L2-Betti numbers

Cited by in corpus (20)

• Approximating L2-invariants, and the Atiyah conjecture
• On a conjecture of Atiyah
• L2-determinant class and approximation of L2-Betti numbers
• Integrality of L2-Betti numbers
• Rational group ring elements with kernels having irrational dimension
• Freely Independent Random Variables with Non-Atomic Distributions
• 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
• Spectral and Combinatorial Aspects of Cayley-Crystals
• The Atiyah conjecture and Artinian rings
• Groups of small homological dimension and the Atiyah Conjecture
• Operator algebras and topology
• Compact Group Automorphisms, Addition Formulas and Fuglede-Kadison Determinants
• The trace on the K-theory of group C*-algebras
• Traces on group C*-algebras, sofic groups and Lück's conjecture
• Some remarks on the entropy for algebraic actions of amenable groups
• Hyperlinearity, essentially free actions and $L^2$-invariants. The sofic property
• The Strong Atiyah conjecture for virtually cocompact special groups