Hattori-Stallings trace and Euler characteristics for groups

Chatterji, Indira; Mislin, Guido

doi:10.1017/cbo9781139107099.007

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The Main Theorem1

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2021

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Cited by 3 publications

(2 citation statements)

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“…To prove Theorem 1.1, one needs the following method of computing the ℓ 2 -Euler characteristic of a group acting on a tree analogous to Chiswell's result [9] for rational Euler characteristic. Proposition 3.1 (Chatterji-Mislin [8]). Let F be a group acting on a tree and let V and E denote sets of representatives of F-orbits of vertices and edges.…”

Section: The Main Theoremmentioning

confidence: 99%

The first -betti number and groups acting on trees

Chatterji¹,

Hughes²,

Kropholler³

2021

Proceedings of the Edinburgh Mathematical Society

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Section: The Main Theoremmentioning

confidence: 99%

The first -betti number and groups acting on trees

Chatterji¹,

Hughes²,

Kropholler³

2021

Proceedings of the Edinburgh Mathematical Society

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“…In order to give a positive answer to his conjecture, Brown considered more general situation as in [6,Theorem 3.1]. In 2000, Chatterji and Mislin [8] conjectured that if G is a group of type F P over C such that the centralizer of every element of finite order in G has finite…”

Section: Introductionmentioning

confidence: 99%

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

Jo¹

2007

J. Math. Soc. Japan

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Hattori-Stallings trace and Euler characteristics for groups

Cited by 3 publications

References 23 publications

The first -betti number and groups acting on trees

The first -betti number and groups acting on trees

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

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

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