The K-theoretic Farrell–Jones conjecture for hyperbolic groups

Bartels, Arthur; Lück, Wolfgang; Reich, Holger

doi:10.1007/s00222-007-0093-7

Cited by 117 publications

(168 citation statements)

References 54 publications

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“…Finally, we mention that Bartels, Lück and Reich have studied a concept closely related to ours in [2]; this idea plays a key role in their proof of the Farrell-Jones conjecture for hyperbolic groups in [3].…”

Section: Introductionmentioning

confidence: 91%

Rokhlin Dimension and C*-Dynamics

Hirshberg

Winter

Zacharias

2015

Commun. Math. Phys.

159

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Abstract:We develop the concept of Rokhlin dimension for integer and for finite group actions on C * -algebras. Our notion generalizes the so-called Rokhlin property, which can be thought of as Rokhlin dimension 0. We show that finite Rokhlin dimension is prevalent and appears in cases in which the Rokhlin property cannot be expected: the property of having finite Rokhlin dimension is generic for automorphisms of Z-stable C * -algebras, where Z denotes the Jiang-Su algebra. Moreover, crossed products by automorphisms with finite Rokhlin dimension preserve the property of having finite nuclear dimension, and under a mild additional hypothesis also preserve Z-stability. In topological dynamics our notion may be interpreted as a topological version of the classical Rokhlin lemma: automorphisms arising from minimal homeomorphisms of finite dimensional compact metrizable spaces always have finite Rokhlin dimension. The latter result has by now been generalized by Szabó to the case of free and aperiodic Z d -actions on compact metrizable and finite dimensional spaces.

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Section: Introductionmentioning

confidence: 91%

Rokhlin Dimension and C*-Dynamics

Hirshberg

Winter

Zacharias

2015

Commun. Math. Phys.

159

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“…Then p −1 (V ) ∼ = F n+1 ⋊V is a hyperbolic group. Thus it satisfies the K-FJCw ( [3], [16]). The result follows from Remark 2.1.…”

Section: The Lower K-theory For H (N)mentioning

confidence: 97%

“…Proof. We will use induction on n. For n = 0, H (0) = F 2 , for which the fibered isomorphism conjecture holds ( [3]). For n = 1, the result in Proposition 3.1.…”

Section: The Lower K-theory For H (N)mentioning

confidence: 99%

On the K-theory of certain extensions of free groups

Metaftsis

Prassidis

2015

Forum Mathematicum

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Since $\operatorname*{Hol}(F_{n})$ embeds into $\operatorname*{Aut}(F_{n+1})$, one can construct inductively the subgroups ${\mathcal{H}}_{(n)}$ of $\operatorname*{Aut}(F_{n+1})$ by setting ${{\mathcal{H}}_{(1)}=\operatorname*{Hol}(F_{2})}$ and ${{\mathcal{H}}_{(n)}=F_{n+1}\rtimes{\mathcal{H}}_{(n-1)}}$. We show that the FJCw holds for ${\mathcal{H}}_{(n)}$. Moreover, we calculate the lower K-theory for the groups ${\mathcal{H}}_{(n)}$.

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“…Conjecture 3.5, even in its much stronger integral version that we are not discussing here, is known to be true for all Gromov hyperbolic groups [BLR08] and all CAT(0)-groups [BL12,Weg12], for example. Theorem 1.3 and Corollary 1.5 have the following immediate consequence.…”

Section: Assembly Maps and Algebraic K-theory Of Tmentioning

confidence: 96%

“…Conjecture 3.1 is known to be true for all Gromov hyperbolic groups [BLR08] and all CAT(0)-groups [BL12], for example. One of the most interesting open cases of Conjecture 3.1 is Thompson's group F : is Wh(F ) = 0?…”

Section: Assembly Maps and Algebraic K-theory Of Tmentioning

confidence: 99%

On Thompson's group T and algebraic K-theory

Geoghegan¹,

Varisco²

Geometric and Cohomological Group Theory

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Using a theorem of Lück-Reich-Rognes-Varisco, we show that the Whitehead group of Thompson's group T is infinitely generated, even when tensored with the rationals. To this end we describe the structure of the centralizers and normalizers of the finite cyclic subgroups of T , via a direct geometric approach based on rotation numbers. This also leads to an explicit computation of the source of the Farrell-Jones assembly map for the rationalized higher algebraic K-theory of the integral group ring of T .

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The K-theoretic Farrell–Jones conjecture for hyperbolic groups

Cited by 117 publications

References 54 publications

Rokhlin Dimension and C*-Dynamics

Rokhlin Dimension and C*-Dynamics

On the K-theory of certain extensions of free groups

On Thompson's group T and algebraic K-theory

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