K-THEORY FOR THE <i>C</i>*-ALGEBRAS OF THE SOLVABLE BAUMSLAG–SOLITAR GROUPS

Pooya, Sanaz; Valette, Alain

doi:10.1017/s0017089517000210

Cited by 9 publications

(8 citation statements)

References 12 publications

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“…, n, under ∂ 1 in the Pimsner-Voiculescu 6-term exact sequence. This generalises Lemma 2 in [11]. Proposition 3.2.…”

Section: Casesupporting

confidence: 75%

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K-theory and K-homology of finite wreath products with free groups

Pooya¹

2019

Illinois J. Math.

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Consider the wreath product Γ = F F n = Fn F F n , with F a finite group and F n the free group on n generators. We study the Baum-Connes conjecture for this group. Our aim is to explicitly describe the Baum-Connes assembly map for F F n . To this end, we compute the topological and the analytical K-groups and exhibit their generators. Moreover, we present a concrete 2-dimensional model for EΓ. As a result of our K-theoretic computations, we obtain that K 0 (C * r (Γ)) is the free abelian group of countable rank with a basis consisting of projections in C * r ( Fn F ) and K 1 (C * r (Γ)) is the free abelian group of rank n with a basis consisting of the unitaries coming from the free group.

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“…, n, under ∂ 1 in the Pimsner-Voiculescu 6-term exact sequence. This generalises Lemma 2 in [11]. Proposition 3.2.…”

Section: Casesupporting

confidence: 75%

“…In this respect, at the very beginning of our way, we try to elucidate the isomorphism for some groups for which the conjecture is satisfied. We have started our investigation in [11] and [3]. This work can be considered a generalisation of the latter.…”

Section: Introductionmentioning

confidence: 99%

K-theory and K-homology of finite wreath products with free groups

Pooya¹

2019

Illinois J. Math.

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“…Hence the diagram above reduces to

\begin{matrix} double-struckZ false({false(M i n (F) false)}^{false(double-struckZ false)} false) & \overset{I d - α_{*}}{⟶} & double-struckZ false({false(M i n (F) false)}^{false(double-struckZ false)} false) & \overset{ι_{*}}{⟶} & K_{0} (C^{*} false(L false)) \\ \partial_{1} ↑ & ↓ \\ K_{1} (C^{*} L) & ⟵ & 0 & ⟵ & 0 . \end{matrix}

By injectivity of

\partial_{1}

and exactness of the sequence

K_{1} false(C^{*} L false) ≃ K e r false(I d - α_{*} false) = Z . p_{F}

, where

p_{F}

denotes the constant map taking the value

p_{F}

double-struckZ

, it corresponds to the class

[1]

of 1 in

K_{0} false(C^{*} B false)

(see Corollary ). Moreover, we know from [, Lemma 2] that

\partial_{1} false[u false] = - false[1 false]

. Hence

K_{1} false(C^{*} L false) = Z . false[u false]

.…”

Section: Lamplighter Groups Over Finite Groupsmentioning

confidence: 99%

“…Let

false[u false] \in K_{1} false(ρ (C^{*} B) ⋊_{α} double-struckZ false)

be the unitary implementing the shift. By [, Lemma 2], we have

\partial [u] = - [1]

and

[1] \neq 0

M_{d^{\infty}} false(double-struckC false)

carries a unital trace. (Slightly more work gives

K_{1} false(ρ (C^{*} B) ⋊_{α} double-struckZ false) = Z false[\frac{1}{d} false]

.)…”

Section: Distinguishing C∗l Up To Isomorphismmentioning

confidence: 99%

K ‐homology and K ‐theory for the lamplighter groups of finite groups

Flores

Pooya

Valette

2017

Proc. London Math. Soc.

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Let F be a finite group. We consider the lamplighter group L=F≀Z over F. We prove that L has a classifying space for proper actions normalE̲L which is a complex of dimension 2. We use this to give an explicit proof of the Baum–Connes conjecture (without coefficients) that states that the assembly map μiL:KiLfalse(E̲0.16emLfalse)→Kifalse(C∗Lfalse)false(i=0,1false) is an isomorphism. Actually, K0false(C∗Lfalse) is free abelian of countable rank, with an explicit basis consisting of projections in C∗L, while K1false(C∗Lfalse) is infinite cyclic, generated by the unitary of C∗L implementing the shift. Finally we show that, for F abelian, the C∗‐algebra C∗L is completely characterized by |F| up to isomorphism.

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“…For 3-dimensional groups, Lück [15] completed this calculation for the semi-direct product Hei 3 (Z) Z 4 of the 3-dimensional integral Heisenberg group with a specific action of the cyclic group Z 4 . Some further computations were completed by Isely [8] for groups of the form Z 2 Z; by Rahm [23] for the class of Bianchi groups; by Pooya and Valette [22] for solvable Baumslag-Solitar groups; and by Flores, Pooya and Valette [6] for lamplighter groups of finite groups.…”

Section: Introductionmentioning

confidence: 99%

Equivariant K-homology for hyperbolic reflection groups

Lafont

Ortiz²,

Rahm³

et al. 2018

The Quarterly Journal of Mathematics

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We compute the equivariant K-homology of the classifying space for proper actions, for cocompact 3-dimensional hyperbolic reflection groups. This coincides with the topological K-theory of the reduced C * -algebra associated to the group, via the Baum-Connes conjecture. We show that, for any such reflection group, the associated K-theory groups are torsion-free. As a result we can promote previous rational computations to integral computations. Our proof relies on a new efficient algebraic criterion for checking torsion-freeness of K-theory groups, which could be applied to many other classes of groups.Date: April 4, 2019. 1 arXiv:1707.05133v2 [math.KT] 3 Apr 2018 2 LAFONT, ORTIZ, RAHM, AND SÁNCHEZ-GARCÍAMain Theorem. Let Γ be a cocompact 3-dimensional hyperbolic reflection group, generated by reflections in the side of a hyperbolic polyhedron P ⊂ H 3 . Then, where the integers cf (Γ), χ(C) can be explicitly computed from the combinatorics of the polyhedron P.Here, cf (Γ) denotes the number of conjugacy classes of elements of finite order in Γ, and χ(C) denotes the Euler characteristic of the Bredon chain complex. By a celebrated result of Andre'ev [2], there is a simple algorithm that inputs a Coxeter group Γ, and decides whether or not there exists a hyperbolic polyhedron P Γ ⊂ H 3 which generates Γ. In particular, given an arbitrary Coxeter group, one can easily verify if it satisfies the hypotheses of our Main Theorem.Note that the lack of torsion in the K-theory is not a property shared by all discrete groups acting on hyperbolic 3-space. For example, 2-torsion occurs in K 0 (C * r (Γ)) whenever Γ is a Bianchi group containing a 2-dihedral subgroup C 2 ×C 2 (see [23]). In fact, the key difficulty in the proof of our Main Theorem lies in showing that these K-theory groups are torsion-free. Some previous integral computations yielded K-theory groups that are torsion-free, though in those papers the torsionfreeness was a consequence of ad-hoc computations. Our second goal is to give a general criterion which explains the lack of torsion, and can be efficiently checked. This allows a systematic, algorithmic approach to the question of whether a Ktheory group is torsion-free.Let us briefly describe the contents of the paper. In Section 2, we provide background material on hyperbolic reflection groups, topological K-theory, and the Baum-Connes Conjecture. We also introduce our main tool, the Atiyah-Hirzebruch type spectral sequence. In Section 3, we use the spectral sequence to show that the K-theory groups we are interested in coincide with the Bredon homology groups H Fin 0 (Γ; R C ) and H Fin 1 (Γ; R C ) respectively. We also explain, using the Γ-action on H 3 , why the homology group H Fin 1 (Γ; R C ) is torsion-free. In contrast, showing that H Fin 0 (Γ; R C ) is torsion-free is much more difficult. In Section 4, we give a geometric proof for this fact in a restricted setting. In Section 5, we give a linear algebraic proof in the general case, inspired by the "representation ring splitting" technique of [23]....

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K-THEORY FOR THE C*-ALGEBRAS OF THE SOLVABLE BAUMSLAG–SOLITAR GROUPS

Cited by 9 publications

References 12 publications

K-theory and K-homology of finite wreath products with free groups

K-theory and K-homology of finite wreath products with free groups

K ‐homology and K ‐theory for the lamplighter groups of finite groups

Equivariant K-homology for hyperbolic reflection groups

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