Asymptotic linearity of the mapping class group and a homological version of the Nielsen–Thurston classification

We study the action of the mapping class group on the integral homology of finite covers of a topological surface. We use the homological representation of the mapping class to construct a faithful infinite-dimensional representation of the mapping class group. We show that this representation detects the Nielsen–Thurston classification of each mapping class. We then discuss some examples that occur in the theory of braid groups and develop an analogous theory for automorphisms of free groups. We close with some open problems.     

Mapping Class Groups of Nonorientable Surfaces

We obtain a finite set of generators for the mapping class group of a nonorientable surface with punctures. We then compute the first homology group of the mapping class group and certain subgroups of it. As an application we prove that the image of a homomorphism from the mapping class group of a nonorientable surface of genus at least nine to the group of real-analytic diffeomorphisms of the circle is either trivial or of order two.     

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.     

Representations of groups with CAT(0) fixed point property

We show that certain representations over fields with positive characteristic of groups having CAT\((0)\) fixed point property \(\mathrm{F}\mathcal {B}_{\widetilde{A}_n}\) have finite image. In particular, we obtain rigidity results for representations of the following groups: the special linear group over \({\mathbb {Z}}\), \({\mathrm{SL}}_k({\mathbb {Z}})\), the special automorphism group of a free group, \(\mathrm{SAut}(F_k)\), the mapping class group of a closed orientable surface, \(\mathrm{Mod}(\Sigma _g)\), and many other groups. In the case of characteristic zero, we show that low dimensional complex representations of groups having CAT\((0)\) fixed point property \(\mathrm{F}\mathcal {B}_{\widetilde{A}_n}\) have finite image if they always have compact closure.     

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.

     

Hölder estimates for the\bar \partial - equation in some domains of finite typein some domains of finite type

We obtain Hölder estimates for the $\bar \partial - equation$ on some domains of finite type in ?ⁿ using proper mapping techniques. The domains considered are domains of finite type in the sense of D’Angelo and are defined by local coordinate expressions satisfying certain algebraic geometric conditions which prevent the existence of complex analytic varieties in the boundary of the domain. Using a proper mapping which is given by the finite type condition and which carries all the information about the intrinsic geometry of the boundary, we transform the finite type points into strongly pseudoconvex ones. At these strongly pseudoconvex points we compute an explicit solution using the Henkin integral formula and we obtain estimates that we are able to pull back to the original domain. We achieve this by exploiting the branching behavior of the proper mapping. We also construct some biholomorphic numerical invariants associated with some of the domains under consideration.     

On finite loops with nilpotent inner mapping groups

We shall show how the nilpotency class of a finite loop Q is determined by the properties of a nilpotent inner mapping group. We also show that a classical result by Baer on the structure of abelian finite capable groups holds for Moufang loops of odd order.     

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.     

有限义性质下一类广义MORAN集的重分形分解

本文在一类广义Moran集上定义了一种弱分离条件－有限交性质,并证明了重分形分解在广泛的一类分形上仍然成立。本文部分的推广了Cawley和Manlain的结果。     

Rigidity of complete manifolds with parallel Cotton tensor

The aim of this paper is to show some rigidity results for complete Riemannian manifolds with parallel Cotton tensor. In particular, we prove that any compact manifold of dimension \(n\ge 3\) with parallel Cotton tensor and positive constant scalar curvature is isometric to a finite quotient of \({\mathbb {S}}^n\) under a pointwise or integral pinching condition. Moreover, a rigidity theorem for stochastically complete manifolds with parallel Cotton tensor is also given. The proofs rely mainly on curvature elliptic estimates and the weak maximum principle.     

On <Emphasis Type="Italic">G</Emphasis>-rigid surfaces

Rigid algebraic varieties form an important class of complex varieties that exhibit interesting geometric phenomena. In this paper we propose a natural extension of rigidity to complex projective varieties with a finite group action (G-varieties) and focus on the first nontrivial case, namely, on G-rigid surfaces that can be represented as desingularizations of Galois coverings of the projective plane with Galois group G. We obtain local and global G-rigidity criteria for these G-surfaces and present several series of such surfaces that are rigid with respect to the action of the deck transformation group.     

A presentation for the mapping class group of the closed non-orientable surface of genus 4

In [B. Szepietowski, A presentation for the mapping class group of a non-orientable surface from the action on the complex of curves, Osaka J. Math. 45 (2008) 283-326] we proposed a method of finding a finite presentation for the mapping class group of a non-orientable surface by using its action on the so called ordered complex of curves. In this paper we use this method to obtain an explicit finite presentation for the mapping class group of the closed non-orientable surface of genus 4. The set of generators in this presentation consists of 5 Dehn twists, 3 crosscap transpositions and one involution, and it can be immediately reduced to the generating set found by Chillingworth [D.R.J. Chillingworth, A finite set of generators for the homeotopy group of a non-orientable surface, Proc. Camb. Phil. Soc. 65 (1969) 409-430].     

Bounding the rank of Hermitian forms and rigidity for CR mappings of hyperquadrics

Using Green’s hyperplane restriction theorem, we prove that the rank of a Hermitian form on the space of holomorphic polynomials is bounded by a constant depending only on the maximum rank of the form restricted to affine manifolds. As an application we prove a rigidity theorem for CR mappings between hyperquadrics in the spirit of the results of Baouendi–Huang and Baouendi–Ebenfelt–Huang. Given a real-analytic CR mapping of a hyperquadric (not equivalent to a sphere) to another hyperquadric $Q(A,B)$ , either the image of the mapping is contained in a complex affine subspace, or $A$ is bounded by a constant depending only on $B$ . Finally, we prove a stability result about existence of nontrivial CR mappings of hyperquadrics. That is, as long as both $A$ and $B$ are sufficiently large and comparable, then there exist CR mappings whose image is not contained in a hyperplane. The rigidity result also extends when mapping to hyperquadrics in infinite dimensional Hilbert-space.     

Immersions of Finite Geometric Type in Euclidean Spaces

In this paper, we introduce the class of hypersurfaces of finitegeometric type. They are defined as the ones that share the basicdifferential topological properties of minimal surfaces of finite totalcurvature. We extend to surfaces in this class the classical theorem ofOsserman on the number of omitted points of the Gauss mapping ofcomplete minimal surfaces of finite total curvature. We give aclassification of the even-dimensional catenoids as the only even-dimensional minimal hypersurfaces of R ⁿ of finite geometric type.     

Finite covers of random 3-manifolds

A 3-manifold is Haken if it contains a topologically essential surface. The Virtual Haken Conjecture posits that every irreducible 3-manifold with infinite fundamental group has a finite cover which is Haken. In this paper, we study random 3-manifolds and their finite covers in an attempt to shed light on this difficult question. In particular, we consider random Heegaard splittings by gluing two handlebodies by the result of a random walk in the mapping class group of a surface. For this model of random 3-manifold, we are able to compute the probabilities that the resulting manifolds have finite covers of particular kinds. Our results contrast with the analogous probabilities for groups coming from random balanced presentations, giving quantitative theorems to the effect that 3-manifold groups have many more finite quotients than random groups. The next natural question is whether these covers have positive betti number. For abelian covers of a fixed type over 3-manifolds of Heegaard genus 2, we show that the probability of positive betti number is 0.In fact, many of these questions boil down to questions about the mapping class group. We are led to consider the action of the mapping class group of a surface Σ on the set of quotients π₁(Σ)→Q. If Q is a simple group, we show that if the genus of Σ is large, then this action is very mixing. In particular, the action factors through the alternating group of each orbit. This is analogous to Goldman’s theorem that the action of the mapping class group on the SU(2) character variety is ergodic. Mathematics Subject Classification (2000) 57M50, 57N10     

Outer automorphism groups of certain 1-relator groups

Grossman first showed that outer automorphism groups of 1-relator groups given by orientable surface groups are residually finite, whence mapping class groups of orientable surfaces are residually finite. Allenby, Kim and Tang showed that outer automorphism groups of cyclically pinched 1-relator groups are residually finite, whence mapping class groups of orientable and non-orientable surfaces are residually finite. In this paper we show that outer automorphism groups of certain conjugacy separable 1-relator groups are residually finite.     

A Bowen type rigidity theorem for non-cocompact hyperbolic groups

We establish a Bowen type rigidity theorem for the fundamental group of a noncompact hyperbolic manifold of finite volume (with dimension at least 3).      

Fast Algorithms for Identification and Comparison of Braids

A method for constructing algorithms solving the word and comparison problems for mapping class groups (in particular, for the braid group) is presented, and a family of one-side invariant orderings on the mapping class group of a surface with boundary is described. A method for constructing comparison algorithms for all finite orderings on the mapping class group of any surface with boundary is described, a fast and simple comparison algorithm for the Dehornoy order on the braid group is presented, examples of normal forms for braid groups are given, and algorithms for finding the forms are indicated. Bibliography: 15 titles.     

Parabolic subgroups in FC-type Artin groups

Parabolic subgroups are the building blocks of Artin groups. This paper extends previous results of Cumplido, Gebhardt, Gonzales-Meneses and Wiest, known only for parabolic subgroups of finite type Artin groups, to parabolic subgroups of FC-type Artin groups. We show that the class of finite type parabolic subgroups is closed under intersection. We also study an analog of the curve complex for mapping class group constructed by Cumplido et al. using parabolic subgroups. We extend the construction of this complex, called the complex of parabolic subgroups, to FC-type Artin groups. We show that this simplicial complex is, in most cases, infinite diameter and conjecture that it is δ-hyperbolic.     

A rigidity theorem for group extensions

We consider the class of discrete groups which arise as fundamental groups of iterated surface fibrations; that is, of complexes obtained from a sequence of fibrations in which all bases and the initial fibre are hyperbolic surfaces. Group theoretically, this corresponds to studying the class of iterated extensions of hyperbolic surface groups. In [4], for the case of a single extension we conjectured and partially established that no group can arise from more than a finite number of such extensions. Here we show that the result holds in complete generality. As remarked in [4], the result has a strong affinity with the rigidity theorems of Parshin [7] and Arakelov [1] for fibred (complex) algebraic surfaces.