The Farrell–Jones conjecture for virtually solvable groups:

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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“…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: 99%

“…In [16], it was shown that CAT(0)-groups satisfy the K-FJCw. That means that finite extensions of CAT(0)-groups satisfy the K-FJCw.…”

Section: Preliminaries and Notationmentioning

confidence: 99%

On the K-theory of certain extensions of free groups

Metaftsis

Prassidis

2015

Forum Mathematicum

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“…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. On the other hand, hyperbolic groups satisfy both conjectures.…”

mentioning

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 virtually solvable groups:

Abstract: Abstract. We prove the K-and L-theoretic Farrell-Jones Conjecture (with coefficients in additive categories) for virtually solvable groups.

Cited by 40 publications

References 23 publications

On the K-theory of certain extensions of free groups

On the K-theory of certain extensions of free groups

K-THEORY AND ACTIONS ON EUCLIDEAN RETRACTS

Twisting L2-invariants with finite-dimensional representations

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