A geometric approach to the lower algebraic K-theory of Fuchsian groups

Berkove, Ethan; Juan-Pineda, Daniel; Pearson, Kimberly

doi:10.1016/s0166-8641(01)00068-2

Cited by 9 publications

(13 citation statements)

References 8 publications

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“…N G (H) ∼ = D ∞ ). This generalizes the main result of [BJPP02] and [BJPP01]. A particular interesting example is the fundamental group of an orientable closed surface of genus at least two.…”

Section: Msupporting

confidence: 83%

The algebraic and topological $K$-theory of the Hilbert modular group

Saldaña¹,

Velásquez

2018

Homology, Homotopy and Applications

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In this paper we provide descriptions of the Whitehead groups with coefficients in a ring for the Hilbert modular group (and its reduced version). We also compute the rational topological K-theory of their reduced C * -algebras. This is done by computing the source of the assembly maps in the Farrell-Jones and the Baum-Connes conjecture respectively. We also construct a model for the classifying space of the Hilbert modular group for the family of virtually cyclic subgroups.

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Section: Msupporting

confidence: 83%

The algebraic and topological $K$-theory of the Hilbert modular group

Saldaña¹,

Velásquez

2018

Homology, Homotopy and Applications

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“…This conjecture has been verified for several groups, for instance, for discrete cocompact subgroups of virtually connected Lie groups by Farrell and Jones [13], for finitely generated Fuchsian groups by Berkove, Juan-Pineda and Pearson [5] and for some mapping class groups by Berkove, Juan-Pineda and Lu in [4].…”

Section: Introductionmentioning

confidence: 82%

The Whitehead group and the lower algebraicK–theory of braid groups on 𝕊²and ℝP²

Juan-Pineda

Millán-López

2010

Algebr. Geom. Topol.

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1887The Whitehead group and the lower algebraic K -theory of braid groups on S 2 and RP of vanishes for all n > 0, z K 0 .Z/ vanishes for all n > 0 except for the cases n D 2; 3 and K i .Z/ vanishes for all i Ä 1. 19A31, 19B28; 55N25 IntroductionAravinda, Farrell, and Roushon in [2] showed that if is the pure braid group on any compact (connected) surface, except for the 2-sphere S 2 and the real projective plane RP 2 , then the Whitehead group Wh./ of vanishes. Later on, in [14], Farrell and Roushon extended this result to the full braid groups. They showed that if is the full braid group on any compact (connected) surface, except for the 2-sphere and the 2-projective plane, then Wh./ also vanishes. A natural question is what happens to the Whitehead group of the pure and full braid groups in the two remaining cases?The main tool to answer this question is the Fibered Isomorphism Conjecture of Farrell and Jones [13]. This conjecture has been verified for several groups, for instance, for discrete cocompact subgroups of virtually connected Lie groups by Farrell and Jones [13], for finitely generated Fuchsian groups by Berkove, Juan-Pineda and Pearson [5] and for some mapping class groups by Berkove, Juan-Pineda and Lu in [4].Let M D S 2 or RP 2 . Let PB n .M / and B n .M / be the pure and the full braid groups on M respectively. In this paper we recall from [18; 19] that PB n .M / and B n .M / satisfy the Farrell-Jones isomorphism conjecture and use this fact to compute the Whitehead group of PB n .M / and the lower algebraic K -groups for the integral group

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“…It is an important fact that when n is prime N Γ i g (Z/n) = C Γ i g (Z/n), i.e., the normalizer and centralizer agree. Therefore, any extension of Z by a prime order cyclic group must be a direct product, proving (2). On the other hand, an extension of Z by a composite order cyclic group will split, yielding Z/n ⋊ Z. Extensions of Z by prime cyclics appear as subgroups of this semidirect product, implying (3).…”

Section: Calculationsmentioning

confidence: 92%