A remark on homomorphisms from right-angled Artin groups to mapping class groups

We study rigidity properties of certain homomorphisms from right-angled Artin groups to mapping class groups. As an application, we show that if $\Gamma ?\mathrm{Map}(S)$ is a subgroup that contains some power of every Dehn twist, then any injective homomorphism $\Gamma \to \mathrm{Map}(S)$ is a restriction of an automorphism of $\mathrm{Map}(S)$.     

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).     

Bordism invariants of the mapping class group

We define new bordism and spin bordism invariants of certain subgroups of the mapping class group of a surface. In particular, they are invariants of the Johnson filtration of the mapping class group. The second and third terms of this filtration are the well-known Torelli group and Johnson subgroup, respectively. We introduce a new representation in terms of spin bordism, and we prove that this single representation contains all of the information given by the Johnson homomorphism, the Birman-Craggs homomorphism, and the Morita homomorphism.     

Sieve methods in group theory II: the mapping class group

We prove that the set of non-pseudo-Anosov elements in the Torelli group is exponentially small. This answers a question of Kowalski (2008).     

The Poisson boundary of the mapping class group

A theory of random walks on the mapping class group and its non-elementary subgroups is developed. We prove convergence of sample paths in the Thurston compactification and show that the space of projective measured foliations with the corresponding harmonic measure can be identified with the Poisson boundary of random walks. The methods are based on an analysis of the asymptotic geometry of Teichmüller space and of the contraction properties of the action of the mapping class group on the Thurston boundary. We prove, in particular, that Teichmüller space is roughly isometric to a graph with uniformly bounded vertex degrees. Using our analysis of the mapping class group action on the Thurston boundary we prove that no non-elementary subgroup of the mapping class group can be a lattice in a higher rank semi-simple Lie group. Oblatum 10-V-1995 & 11-IX-1995     

Realization of the mapping class group by homeomorphisms

In this paper, we show that the mapping class group of a closed surface can not be geometrically realized as a group of homeomorphisms of that surface. More precisely, let denote the standard projection of the group of homeomorphisms to the mapping class group of a closed surface M of genus g>5. We show that there is no homomorphism , such that is the identity. This answers a question by Thurston (see [11]). Mathematics Subject Classification (2000)  Primary 20H10, 37F30     

On the homotopy of the stable mapping class group

By considering all surfaces and their mapping class groups at once, it is shown that the classifying space of the stable mapping class group after plus construction, BΓ_∞ ⁺, has the homotopy type of an infinite loop space. The main new tool is a generalized group completion theorem for simplicial categories. The first deloop of BΓ_∞ ⁺ coincides with that of Miller [M] induced by the pairs of pants multiplication. The classical representation of the mapping class group onto Siegel's modular group is shown to induce a map of infinite loop spaces from BΓ_∞ ⁺ to K-theory. It is then a direct consequence of a theorem by Charney and Cohen [CC] that there is a space Y such that BΓ_∞ ⁺≃Im J _(1/2)×Y, where Im J _(1/2) is the image of J localized away from the prime 2. Oblatum 23-X-1995 &19-XI-1996     

Computing class fields via the Artin map

Based on an explicit representation of the Artin map for Kummer extensions, we present a method to compute arbitrary class fields. As in the proofs of the existence theorem, the problem is first reduced to the case where the field contains sufficiently many roots of unity. Using Kummer theory and an explicit version of the Artin reciprocity law we show how to compute class fields in this case. We conclude with several examples.

     

A presentation of mapping class groups in terms of Artin groups and geometric monodromy of singularities

From Wajnryb's presentation, we extract a simple presentation of the mapping class group of the genus g surface as a quotient of an Artin group by simple relations among the centers of sub-Artin groups. Topological meanings are given by using deformation of simple singularities. Received: 22 January 1998 / in final form: 16 February 1999     

Actions and irreducible representations of the mapping class group

Let G be a countable discrete group. Call two subgroups and of G commensurable if has finite index in both and . We say that an action of G on a discrete set X has noncommensurable stabilizers if the stabilizers of any two distinct points of X are not commensurable. We prove in this paper that the action of the map ping class group on the complex of curves has noncommensurable stabilizers. Following a method due to Burger and de la Harpe, this action leads to constructions of irreducible unitary representations of the mapping class group. Received: 26 July 1999 / Revised version: 14 May 2001 / Published online: 19 October 2001     

Generating the surface mapping class group by two elements

Wajnryb proved in 1996 that the mapping class group of an orientable surface is generated by two elements. We prove that one of these generators can be taken as a Dehn twist. We also prove that the extended mapping class group is generated by two elements, again one of which is a Dehn twist. Another result we prove is that the mapping class groups are also generated by two elements of finite order.

     

Signature of relations in mapping class groups and non-holomorphic Lefschetz fibrations

We introduce the notion of signature for relations in mapping class groups and show that the signature of a Lefschetz fibration over the 2-sphere is the sum of the signatures for basic relations contained in its monodromy. Combining explicit calculations of the signature cocycle with a technique of substituting positive relations, we give some new examples of non-holomorphic Lefschetz fibrations of genus and which violate slope bounds for non-hyperelliptic fibrations on algebraic surfaces of general type.

     

Big Heegaard distance implies finite mapping class group

We show that if M is a closed three manifold with a Heegaard splitting with sufficiently big Heegaard distance then the subgroup of the mapping class group of the Heegaard surface, whose elements extend to both handlebodies is finite. As a corollary, this implies that under the same hypothesis, the mapping class group of M is finite.     

Right-angled Artin groups and a generalized isomorphism problem for finitely generated subgroups of mapping class groups

Consider the mapping class group Mod _g,p of a surface ?? _g,p of genus g with p punctures, and a finite collection {f₁, . . . , f_k} of mapping classes, each of which is either a Dehn twist about a simple closed curve or a pseudo-Anosov homeomorphism supported on a connected subsurface. In this paper we prove that for all sufficiently large N, the mapping classes ${\{f_1^N,\ldots,f_k^N\}}$ generate a right-angled Artin group. The right-angled Artin group which they generate can be determined from the combinatorial topology of the mapping classes themselves. When {f₁, . . . , f_k} are arbitrary mapping classes, we show that sufficiently large powers of these mapping classes generate a group which embeds in a right-angled Artin group in a controlled way. We establish some analogous results for real and complex hyperbolic manifolds. We also discuss the unsolvability of the isomorphism problem for finitely generated subgroups of Mod _g,p , and recover the fact that the isomorphism problem for right-angled Artin groups is solvable. We thus characterize the isomorphism type of many naturally occurring subgroups of Mod _g,p .