On vanishing of $L^{2}$-Betti numbers for groups

Jo, Jang Hyun

doi:10.2969/jmsj/05941031

Cited by 2 publications

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“…[8, Lemma 2.1]). Thus every good group over Q or C in the sense of [6] is L (2) -good over Q or C. It is known from [8, Theorems 4.2 and 4.4] and [10] that there are many groups satisfying the condition (c) over C in Definition 8. Note also that if a group G admits a cocompact EG which is the classifying space for proper G-action, then G has only finitely many elements of finite order up to conjugacy (cf.…”

Section: Theorem 7 Let X Be a Finite Dimensional Virtually Free G-cwmentioning

confidence: 98%

Equivariant $L^2$-Euler characteristics of $G\textrm{-}CW$-complexes

2017

Arkiv För Matematik

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Abstract.We show that if X is a cocompact G-CW -complex such that each isotropy subgroup Gσ is L (2) -good over an arbitrary commutative ring k, then X satisfies some fixed-point formula which is an L (2) -analogue of Brown's formula in 1982. Using this result we present a fixed point formula for a cocompact proper G-CW -complex which relates the equivariant L (2) -Euler characteristic of a fixed point CW -complex X s and the Euler characteristic of X/G. As corollaries, we prove Atiyah's theorem in 1976, Akita's formula in 1999 and a result of Chatterji-Mislin in 2009. We also show that if X is a free G-CW -complex such that C * (X) is chain homotopy equivalent to a chain complex of finitely generated projective Zπ 1 (X)-modules of finite length and X satisfies some fixed-point formula over Q or C which is an L (2) -analogue of Brown's formula, then χ(X/G)=χ (2) (X). As an application, we prove that the weak Bass conjecture holds for any finitely presented group G satisfying the following condition: for any finitely dominated CW -complex Y with π 1 (Y )=G, Y satisfies some fixed-point formula over Q or C which is an L (2) -analogue of Brown's formula.

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Section: Theorem 7 Let X Be a Finite Dimensional Virtually Free G-cwmentioning

confidence: 98%