L2-determinant class and approximation of L2-Betti numbers

Abstract

A standing conjecture in L2-cohomology is that every finite CW-complex X is of L2-determinant class. In this paper, we prove this whenever the fundamental group belongs to a large class of groups containing e.g. all extensions of residually finite groups with amenable quotients, all residually amenable groups and free products of these. If, in addition, X is L2-acyclic, we also prove that the L2-determinant is a homotopy invariant. Even in the known cases, our proof of homotopy invariance is much shorter and easier than the previous ones. Under suitable conditions we give new approximation formulas for L2-Betti numbers. Errata are added, rectifying some unproved statements about "amenable extension": throughout, amenable extensions should be extensions with \emph{normal} subgroups.

amsLaTeX2e, 26 pages; v2: Errata are added, rectifying some unproved statements about "amenable extension"

References in corpus (6)

• Approximating L2-invariants, and the Atiyah conjecture
• L2-determinant class and approximation of L2-Betti numbers
• Integrality of L2-Betti numbers
• L^2-torsion of hyperbolic manifolds of finite volume
• On the homotopy invariance of L^2 torsion for covering spaces
• Residual Amenability and the Approximation of L^2-invariants

Cited by in corpus (40)

• 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
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• Closed manifolds with transcendental L2-Betti numbers
• Integrated density of states and Wegner estimates for random Schrödinger Operators
• The strong Atiyah and Lück approximation conjectures for one-relator groups
• Mahler measures and Fuglede--Kadison determinants
• Spectral and Combinatorial Aspects of Cayley-Crystals
• Approximating L^2-signatures by their compact analogues
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• $L^2$-Alexander invariant for knots
• Survey on approximating L^2-invariants by their classical counterparts: Betti numbers, torsion invariants and homological growth
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• Operator algebras and topology
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• Compact Group Automorphisms, Addition Formulas and Fuglede-Kadison Determinants
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• The growth of Betti numbers and approximation theorems
• Traces on group C*-algebras, sofic groups and Lück's conjecture
• L2-invariants of nonuniform lattices in semisimple Lie groups
• Growth of Betti Numbers
• Homoclinic points, atoral polynomials, and periodic points of algebraic Z^d-actions
• Sofic groups and diophantine approximation
• Lehmer's Problem for arbitrary groups
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• Residual properties of free products
• Equivariant Benjamini-Schramm Convergence of Simplicial Complexes and ${\ell }^2$-Multiplicities
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• Expansive algebraic actions of discrete residually finite amenable groups and their entropy
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