The isomorphism conjecture for solvable groups in Waldhausen’s A-theory

Farrell, F. T.; Wu, Xiaolei

doi:10.1142/s1793525319500195

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Introduction2

Remark 1 Osajda and Huang Have Independently Obtained A Proo...1

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2019

2021

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Cited by 3 publications

(3 citation statements)

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“…Remark 2. Note that our results also hold for the Farrell-Jones Conjecture in A-theory since all the results and inheritance properties we used in our proof also holds for A-theory [9,10,14].…”

Section: Remark 1 Osajda and Huang Have Independently Obtained A Proo...supporting

confidence: 69%

The Farrell-Jones Conjecture for normally poly-free groups

Brück

Kielak

2019

Preprint

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Section: Remark 1 Osajda and Huang Have Independently Obtained A Proo...supporting

confidence: 69%

The Farrell-Jones Conjecture for normally poly-free groups

Brück

Kielak

2019

Preprint

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“…As explained in [, Section 3], the analogous statements of Theorem and Corollaries and for (topological, PL or smooth) Whitehead spectra and pseudoisotopy spectra also hold true. Remark We have been informed that Thomas Farrell and Xiaolei Wu have independently obtained a proof of Theorem .…”

Section: Introductionmentioning

confidence: 99%

“…Remark 1.4. We have been informed that Thomas Farrell and Xiaolei Wu [6] have independently obtained a proof of Theorem 1.1.…”

Section: Introductionmentioning

confidence: 99%

The A‐theoretic Farrell–Jones conjecture for virtually solvable groups

Kasprowski

Ullmann

Wegner³

et al. 2017

Bulletin of London Math Soc

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We prove the A-theoretic Farrell-Jones conjecture for virtually solvable groups. As a corollary, we obtain that the conjecture holds for S-arithmetic groups and lattices in almost connected Lie groups.Using this, we can adapt previous work by Rüping [9] and Kammeyer, Lück and Rüping [7] to A-theory:Corollary 1.2. The A-theoretic Farrell-Jones conjecture with coefficients and finite wreath products holds for subgroups of GL n (Q) or GL n (F (t)), where F is a finite field.In particular, the conjecture holds for S-arithmetic groups.Proof. The proof works as the one of [9, Theorem 8.13]: Since the conjecture is inherited under directed colimits [4, Theorem 1.1(ii)], it suffices to consider linear groups over localizations at finitely many primes. Then [9, Proposition 2.2] together with [4, Corollary 6.6]

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The Farrell–Jones conjecture for normally poly-free groups

Brück

Kielak

2021

Proc. Amer. Math. Soc.

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We construct explicit finite-dimensional orthogonal representations πN of SLN (Z) for N ∈ {3, 4} all of whose invariant vectors are trivial, and such that H N −1 (SLN (Z), πN ) is non-trivial. This implies that for N as above, the group SLN (Z) does not have property (TN−1) of Bader-Sauer and therefore is not (N − 1)-Kazhdan in the sense of De Chiffre-Glebsky-Lubotzky-Thom, both being higher versions of Kazhdan's property T .

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The isomorphism conjecture for solvable groups in Waldhausen’s A-theory

Abstract: Abstract. We prove the A-theoretic Isomorphism Conjecture with coefficients and finite wreath products for solvable groups.

Cited by 3 publications

References 16 publications

The Farrell-Jones Conjecture for normally poly-free groups

The Farrell-Jones Conjecture for normally poly-free groups

The A‐theoretic Farrell–Jones conjecture for virtually solvable groups

The Farrell–Jones conjecture for normally poly-free groups

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