Non-positive curvature and complexity for finitely presented groups

Bridson, Martin R.

doi:10.4171/022-2/46

Cited by 13 publications

(14 citation statements)

References 84 publications

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“…We mention that the Main Theorem implies that the K-theoretic Farrell-Jones Conjecture with coefficients in any ring R holds not only for hyperbolic groups but for instance for any group which occurs as a subgroup of a finite product of hyperbolic groups and for any directed colimit of hyperbolic groups (with not necessarily injective structure maps). Such groups can be very wild and can have exotic properties (see Bridson [13] and Gromov [36]). This follows from some general inheritance properties.…”

Section: Some Explanations Are In Ordermentioning

confidence: 99%

The K-theoretic Farrell–Jones conjecture for hyperbolic groups

2007

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Section: Some Explanations Are In Ordermentioning

confidence: 99%

The K-theoretic Farrell–Jones conjecture for hyperbolic groups

2007

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“…Let FJ ≤1 be the class of groups which satisfy the L-theoretic Farrell-Jones Conjecture with additive categories as coefficients and the K-theoretic Farrell-Jones Conjecture with additive categories as coefficients up to degree one. In view of the results above, these classes contain many groups which lie in the region Hic Abundant Leones in Martin Bridson's universe of groups (see [15]). Theorem 4.1 and Theorem 4.3 (iv) imply that directed colimits of hyperbolic groups belong to FJ .…”

Section: Proof See [4 Lemma 23]mentioning

confidence: 96%

K- and L-theory of Group Rings

Lück

2011

Proceedings of the International Congress of Mathematicians 2010 (ICM 2010)

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Abstract. This article will explore the K-and L-theory of group rings and their applications to algebra, geometry and topology. The Farrell-Jones Conjecture characterizes K-and L-theory groups. It has many implications, including the Borel and Novikov Conjectures for topological rigidity. Its current status, and many of its consequences are surveyed. Mathematics Subject Classification (2000). Primary 18F25; Secondary 57XX.Keywords. K-and L-theory, group rings, Farrell-Jones Conjecture, topological rigidity. IntroductionThe algebraic K-and L-theory of group rings -K n (RG) and L n (RG) for a ring R and a group G -are highly significant, but are very hard to compute when G is infinite. The main ingredient for their analysis is the Farrell-Jones Conjecture. It identifies them with certain equivariant homology theories evaluated on the classifying space for the family of virtually cyclic subgroups of G. Roughly speaking, the Farrell-Jones Conjecture predicts that one can compute the values of these K-and L-groups for RG if one understands all of the values for RH, where H runs through the virtually cyclic subgroups of G.Why is the Farrell-Jones Conjecture so important? One reason is that it plays an important role in the classification and geometry of manifolds. A second reason is that it implies a variety of well-known conjectures, such as the ones due to Bass, Borel, Kaplansky and Novikov. (These conjectures are explained in Section 1.) There are many groups for which these conjectures were previously unknown but are now consequences of the proof that they satisfy the Farrell-Jones Conjecture. A third reason is that most of the explicit computations of K-and L-theory of group rings for infinite groups are based on the Farrell-Jones Conjecture, since it identifies them with equivariant homology groups which are more accessible via standard tools from algebraic topology and geometry (see Section 5).The rather complicated general formulation of the Farrell-Jones Conjecture is given in Section 3. The much easier, but already very interesting, special case of a torsionfree group is discussed in Section 2. In this situation the K-and L-groups are identified with certain homology theories applied to the classifying space BG. * The work was financially supported by the Leibniz-Preis of the author. The author wishes to thank several members and guests of the topology group in Münster for helpful comments.

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“…Most of these results rely on the template described in [5]: given a finitely presented group of interest, Q, one uses a variant of the Rips construction [30] to construct an epimorphism p : Γ → Q with finitely generated kernel, where Γ is hyperbolic; one then forms the fibre product P = {(x, y) | p(x) = p(y)} < Γ × Γ. In general, P will be finitely generated but not finitely presented.…”

Section: The Conjugacy and Membership Problemsmentioning

confidence: 99%

On the subgroups of right-angled Artin groups and mapping class groups

Bridson¹

2013

Mathematical Research Letters

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Abstract. There exist right-angled Artin groups A such that the isomorphism problem for finitely presented subgroups of A is unsolvable and for certain finitely presented subgroups the conjugacy and membership problems are unsolvable. It follows that if S is a surface of finite type and the genus of S is sufficiently large, then the corresponding decision problems for the mapping class group Mod(S) are unsolvable. Every virtually special group embeds in the mapping class group of infinitely many closed surfaces. Examples are given of finitely presented subgroups of mapping class groups that have infinitely many conjugacy classes of torsion elements. A finitely presented subgroup of a mapping class group can have an exponential Dehn function.

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Non-positive curvature and complexity for finitely presented groups

Cited by 13 publications

References 84 publications

The K-theoretic Farrell–Jones conjecture for hyperbolic groups

The K-theoretic Farrell–Jones conjecture for hyperbolic groups

K- and L-theory of Group Rings

On the subgroups of right-angled Artin groups and mapping class groups

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