Periodicity of the punctured mapping class group

The main result is that the punctured mapping class group Γ_gⁱ (i≥1, g≥1) has periodic cohomology; furthermore, the period is always 2. We present a proof which involves the Yagita invariant and the Chern class of the representation of a subgroup in Γ_gⁱ (i≥1, g≥1). Using the main result, we can calculate the p-torsion of the Farrell cohomology for some special values of g and i. To do this, we extend the definition of the fixed point data as well as the conjugation theorem known for the case Γ_g⁰ to the case Γ_gⁱ (i≥1, g≥1).     

K-theory of toeplitzC *-algebras on Lie spheres

In this paper we compute theK-groups of theC ^*-algebra of Toeplitz operators on the Lie spheres. As a corollary we get an index theorem for Toeplitz operators with matricial symbols analogous to the index theorem of Berger, Coburn and Koranyi for Toeplitz operators with scalar valued symbols.     

Automorphisms of symmetric groups are toy examples for mapping class groups

We give a new proof establishing the automorphism groups of the symmetric groups inspired by the analogous result of Ivanov for the extended mapping class group. As a key tool, we consider the actions on the Kneser graphs.     

The K-theory of p-compact groups

In this paper, we show that the p-adic K-theory of a connected p-compact is the ring of invariants of the Weyl group action on the K-theory of a maximal torus. We apply this result to show that a connected finite loop space admits a maximal torus if and only if its complex K-theory is -isomorphic to the K-theory of some BG, where G is a compact connected Lie group. Received: November 9, 1996     

Separable subgroups of mapping class groups

We investigate separability questions for the mapping class group of a surface. While this group is not subgroup separable in general, we prove a large family of interesting subgroups are separable. This includes many classically studied subgroups such as solvable subgroups, Heegaard and Handlebody groups, geometric subgroups, and all the terms in the Johnson filtration.     

Higher algebraic K-theory for actions of diagonalizable groups

We study the K-theory of actions of diagonalizable group schemes on noetherian regular separated algebraic spaces: our main result shows how to reconstruct the K-theory ring of such an action from the K-theory rings of the loci where the stabilizers have constant dimension. We apply this to the calculation of the equivariant K-theory of toric varieties, and give conditions under which the Merkurjev spectral sequence degenerates, so that the equivariant K-theory ring determines the ordinary K-theory ring. We also prove a very refined localization theorem for actions of this type. Mathematics Subject Classification (2000) 19E08, 14L30     

Simplicial groups that are models for algebraic K-theory

In this paper, we present and compare some simplicial groups, functorially associated to a ring R, whose homotopy groups are Quillens K-groups of R. The first such simplicial group is the group (NQPR), where is the loop space construction of Clemens Berger, applied to the simplicial set NQPR (the nerve of Quillens category QPR). The second is a subgroup GR of the simplicial group (NQPR). This second group is compared to Kans construction [12] of a loop group for a connected simplicial set, and shown to be isomorphic to it as a simplicial group. Other simplicial groups that are models for algebraic K-theory are also presented; in particular, the subgroup G(s.PR) of (s.PR); here, s.PR is Waldhausens simplicial set [25], [26]. We initially give an exposition of Bergers construction in general; then, we present the construction of GR and a summary of Kans construction. Next, we point out that GR is an infinite loop object in the category of simplicial groups, and draw some corollaries. We then compare directly the homotopy groups thus constructed with the classical K-theory in degrees 0 and 1. The final section compares various models.     

The K-theory of free quantum groups

In this paper we study the $K$ -theory of free quantum groups in the sense of Wang and Van Daele, more precisely, of free products of free unitary and free orthogonal quantum groups. We show that these quantum groups are $K$ -amenable and establish an analogue of the Pimsner–Voiculescu exact sequence. As a consequence, we obtain in particular an explicit computation of the $K$ -theory of free quantum groups. Our approach relies on a generalization of methods from the Baum–Connes conjecture to the framework of discrete quantum groups. This is based on the categorical reformulation of the Baum–Connes conjecture developed by Meyer and Nest. As a main result we show that free quantum groups have a $\gamma $ -element and that $\gamma = 1$ . As an important ingredient in the proof we adapt the Dirac-dual Dirac method for groups acting on trees to the quantum case. We use this to extend some permanence properties of the Baum–Connes conjecture to our setting.     

Zero entropy subgroups of mapping class groups

Given a group action on a surface with a finite invariant set we investigate how the algebraic properties of the induced group of permutations of that set affects the dynamical properties of the group. Our main result shows that in many circumstances if the induced permutation group is not solvable then among the homeomorphisms in the group there must be one with a pseudo-Anosov component. We formulate this in terms of the mapping class group relative to the finite set and show the stronger result that in many circumstances (e.g. if the surface has boundary) if this mapping class group has no elements with pseudo-Anosov components then it is itself solvable.     

Effective generation of right-angled artin groups in mapping class groups

Geometriae Dedicata - We show that given a collection $$X={f_1$$ , ..., $$f_m}$$ of pure mapping classes on a surface S, there is an explicit constant N, depending only on X, such that their Nth...     

Geometry of the mapping class groups I: Boundary amenability

We construct a geometric model for the mapping class group of a non-exceptional oriented surface S of genus g with k punctures and use it to show that the action of on the compact metrizable Hausdorff space of complete geodesic laminations for S is topologically amenable. As a consequence, the Novikov higher signature conjecture holds for every subgroup of .     

Some computations of stable twisted homology for mapping class groups

Abstract

In this paper, we deal with stable homology computations with twisted coefficients for mapping class groups of surfaces and of 3-manifolds, automorphism groups of free groups with boundaries and automorphism groups of certain right-angled Artin groups. On the one hand, the computations are led using semidirect product structures arising naturally from these groups. On the other hand, we compute the stable homology with twisted coefficients by FI-modules. This notably uses a decomposition result of the stable homology with twisted coefficients for pre-braided monoidal categories proved in this paper.

Communicated by Jason P. Bell     

Homological stability for the mapping class groups of non-orientable surfaces

We prove that the homology of the mapping class groups of non-orientable surfaces stabilizes with the genus of the surface. Combining our result with recent work of Madsen and Weiss, we obtain that the classifying space of the stable mapping class group of non-orientable surfaces, up to homology isomorphism, is the infinite loop space of a Thom spectrum built from the canonical bundle over the Grassmannians of 2-planes in ℝ ⁿ⁺². In particular, we show that the stable rational cohomology is a polynomial algebra on generators in degrees 4i – this is the non-oriented analogue of the Mumford conjecture.     

Commensurability of geometric subgroups of mapping class groups

Let M be an orientable surface with punctures and/or boundary components. Paris and Rolfsen (J Reine Angew Math 521:47–83, 2000) studied “geometric subgroups” of the mapping class group of M, that is subgroups corresponding to inclusions of connected subsurfaces. In the present paper we extend this analysis to disconnected subsurfaces and to the nonorientable case. We characterise the subsurfaces which lead to virtually abelian geometric subgroups. We provide algebraic and geometric conditions under which two geometric subgroups are commensurable. We also describe the commensurator of a geometric subgroup in terms of the stabiliser of the underlying subsurface. Finally, following the work of Paris (Math Ann 322:301–315, 2002), we show some applications of our analysis to the theory of irreducible unitary representations of mapping class groups.     

Frattini and related subgroups of mapping class groups

Let Γ_g,b denote the orientation-preserving mapping class group of a closed orientable surface of genus g with b punctures. For a group G let Φ_f(G) denote the intersection of all maximal subgroups of finite index in G. Motivated by a question of Ivanov as to whether Φ_f(G) is nilpotent when G is a finitely generated subgroup of Γ_g,b, in this paper we compute Φ_f(G) for certain subgroups of Γ_g,b. In particular, we answer Ivanov’s question in the affirmative for these subgroups of Γg,b.