The Farrell–Jones conjecture for arbitrary lattices in virtually connected Lie groups

Kammeyer, Holger; Lück, Wolfgang; Rüping, Henrik

doi:10.2140/gt.2016.20.1275

Cited by 32 publications

(26 citation statements)

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“…This result has already been used in several papers, for example, [15,16,21]. The Farrell-Jones conjecture for virtually solvable groups has been studied by several mathematicians.…”

Section: Introductionmentioning

confidence: 88%

“…Let G be a virtually solvable group. Then G satisfies the K-and L-theoretic Farrell-Jones conjecture (with coefficients in additive categories) with respect to the family of virtually cyclic subgroups.This result has already been used in several papers, for example, [15,16,21]. The Farrell-Jones conjecture for virtually solvable groups has been studied by several mathematicians.…”

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confidence: 88%

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The Farrell–Jones conjecture for virtually solvable groups:

Wegner¹

2015

J Topology

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“…This result has already been used in several papers, for example, [15,16,21]. The Farrell-Jones conjecture for virtually solvable groups has been studied by several mathematicians.…”

Section: Introductionmentioning

confidence: 88%

mentioning

confidence: 88%

The Farrell–Jones conjecture for virtually solvable groups:

Wegner¹

2015

J Topology

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“…In this case the parabolic subgroups are slightly bigger, in particular the induction step (on n) here uses that the Farrell-Jones Conjecture holds for all solvable groups. Using inheritance properties and building on these results the Farrell-Jones Conjecture has been verified for all subgroups of GL n (Q) [71] and all lattices in virtually connected Lie groups [44].…”

Section: Covers At Infinitymentioning

confidence: 98%

“…The classes of groups for which the two conjectures are known differ. For example, by work of Kammeyer-Lück-Rüping [44] all lattice in Lie groups satisfy the Farrell-Jones Conjecture; despite Lafforgues [51] positive results for many property T groups, the Baum-Connes Conjecture is still a challenge for SL 3 (Z). Wegner [77] proved the Farrell-Jones Conjecture for all solvable groups, but the case of amenable (or just elementary amenable) groups is open; in contrast Higson-Kasparov [41] proved the Baum-Connes Conjecture for all a-T-menable groups, a class of groups that contains all amenable groups.…”

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confidence: 99%

K-THEORY AND ACTIONS ON EUCLIDEAN RETRACTS

Bartels¹

2019

Proceedings of the International Congress of Mathematicians (ICM 2018)

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This note surveys axiomatic results for the Farrell-Jones Conjecture in terms of actions on Euclidean retracts and applications of these to GLn(Z), relative hyperbolic groups and mapping class groups. IntroductionMotivated by surgery theory Hsiang [43] made a number of influential conjectures about the K-theory of integral group rings Z[G] for torsion free groups G. These conjectures often have direct implications for the classification theory of manifolds of dimension ≥ 5. A good example is the following. An h-cobordism is a compact manifold W that has two boundary components M 0 and M 1 such that both inclusions M i → W are homotopy equivalences. The Whitehead group Wh(G) is the quotient of K 1 (Z[G]) by the subgroup generated by the canonical units ±g, g ∈ G. Associated to an h-cobordism is an invariant, the Whitehead torsion, in Wh(G), where G is the fundamental group of W . A consequence of the s-cobordism theorem is that for dim W ≥ 6, an h-cobordism W is trivial (i.e., isomorphic to a product M 0 ×[0, 1]) iff its Whitehead torsion vanishes. Hsiang conjectured that for G torsion free Wh(G) = 0, and thus that in many cases h-cobordisms are products.The Borel conjecture asserts that closed aspherical manifolds are topologically rigid, i.e., that any homotopy equivalence to another closed manifold is homotopic to a homeomorphism. The last step in proofs of instances of this conjecture via surgery theory uses a vanishing result for Wh(G) to conclude that an h-cobordism is a product and that therefore the two boundary components are homeomorphic.Farrell-Jones [28] pioneered a method of using the geodesic flow on non-positively curved manifolds to study these conjectures. This created a beautiful connection between K-theory and dynamics that led Farrell-Jones [30], among many other results, to a proof of the Borel Conjecture for closed Riemannian manifolds of nonpositive curvature of dimension ≥ 5. Moreover, Farrell-Jones [29] formulated (and proved many instances of) a conjecture about the structure of the algebraic Ktheory (and L-theory) of group rings, even in the presence of torsion in the group. Roughly, the Farrell-Jones Conjecture states that the main building blocks for the K-theory of Z[G] is the K-theory of Z[V ] where V varies of the family of virtually cyclic subgroups of G. It implies a number of other conjectures, among them Hsiang's conjectures, the Borel Conjecture in dimension ≥ 5, the Novikov Conjecture on the homotopy invariant of higher signatures, Kaplansky's conjecture about idempotents in group rings, see [54] for a summary of these and other applications.My goal in this note is twofold. The first goal is to explain a condition formulated in terms of existence of certain actions of G on Euclidean retracts that implies the Farrell-Jones Conjecture for G. This condition was developed in joint work with Lück and Reich [8,12] where the connection between K-theory and dynamics has

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“…The reader should have in mind that it is known for a large class of groups, e.g., hyperbolic groups, CAT(0)-groups, solvable groups, lattices in almost connected Lie groups, fundamental groups of 3-manifolds and passes to subgroups, finite direct products, free products, and colimits of directed systems of groups (with arbitrary structure maps). For more information we refer for instance to [1,2,3,14,25,41,51]. Theorem 6.7 (Basic properties of the V -twisted L 2 -torsion for finite free G-CW -complexes).…”

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confidence: 99%

Twisting L2-invariants with finite-dimensional representations

Lueck

2018

J. Topol. Anal.

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Abstract. We investigate how one can twist L 2 -invariants such as L 2 -Betti numbers and L 2 -torsion with finite-dimensional representations. As a special case we assign to the universal covering X of a finite connected CW -complex X together with an element φ ∈ H 1 (X; R) a φ-twisted L 2 -torsion function R >0 → R, provided that the fundamental group of X is residually finite and X is L 2 -acyclic.

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The Farrell–Jones conjecture for arbitrary lattices in virtually connected Lie groups

Abstract: We prove the K-and the L-theoretic Farrell-Jones conjecture with coefficients in additive categories and with finite wreath products for arbitrary lattices in virtually connected Lie groups.

Cited by 32 publications

References 30 publications

The Farrell–Jones conjecture for virtually solvable groups:

The Farrell–Jones conjecture for virtually solvable groups:

K-THEORY AND ACTIONS ON EUCLIDEAN RETRACTS

Twisting L2-invariants with finite-dimensional representations

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