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     

A Birman exact sequence for Aut(Fn)

The Birman exact sequence describes the effect on the mapping class group of a surface with boundary of gluing discs to the boundary components. We construct an analogous exact sequence for the automorphism group of a free group. For the mapping class group, the kernel of the Birman exact sequence is a surface braid group. We prove that in the context of the automorphism group of a free group, the natural kernel is finitely generated. However, it is not finitely presentable; indeed, we prove that its second rational homology group has infinite rank by constructing an explicit infinite collection of linearly independent abelian cycles. We also determine the abelianization of our kernel and build a simple infinite presentation for it. The keys to many of our proofs are several new generalizations of the Johnson homomorphisms.     

Atoroidal Surface Bundles Over Surfaces

A surface-by-surface group is an extension of a non-trivial orientable closed surface group by another such group. It is an open question as to whether every such group contains a free abelian subgroup of rank 2. We show that, for given base and fibre genera, all but finitely many isomorphism classes of surface-by-surface group contain such an abelian subgroup. This can be rephrased in terms of atoroidal surface bundles over surfaces, or in terms of purely loxodromic surface subgroups of the mapping class groups.     

The multiplicity of abelian covers of splice quotient singularities

From a resolution graph with certain conditions, Neumann and Wahl constructed an equisingular family of surface singularities called splice quotients. For this class some fundamental analytic invariants have been computed from their resolution graph. In this paper we give a method to compute the multiplicity of an abelian covering of a splice quotient from its resolution graph and the Galois group.     

Infinite groups with large balls of torsion elements and small entropy

We exhibit infinite, solvable, virtually abelian groups with a fixed number of generators, having arbitrarily large balls consisting of torsion elements. We also provide a sequence of 3-generator non-virtually nilpotent polycyclic groups of algebraic entropy tending to zero. All these examples are obtained by taking appropriate quotients of finitely presented groups mapping onto the first Grigorchuk group. Received: 3 August 2005     

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.     

Special ideals in partial abelian monoids

Under some conditions, the special congruences of partial abelian monoid are those induced by the special ideals, and a class of special ideals of partial abelian monoid has some upper and lower bound properties.     

An -module structure of the cokernel of the second Johnson homomorphism

The Johnson homomorphisms τ_k (k1) give Abelian quotients of a series of certain subgroups of the mapping class group of a surface. Morita determined the rational image of the second Johnson homomorphism τ₂. In this paper, we study the structure of the torsion part of the cokernel of τ₂. First, we determine the rank of the cokernel over . Although we do it first by computing explicitly, later we improve the proof, using the Birman–Craggs homomorphism, obtained by the classical Rohlin invariant of homology 3-spheres. Since τ₂ is equivariant with respect to the action of the mapping class group, Im τ₂ is -invariant and hence acts on the cokernel. Moreover, computing this action explicitly, we show that the action reduces to that of the finite symplectic group .     

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.     

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.     

The global dimension of polynomial categories in partially commuting variables

We study the global dimension of the category of objects of an abelian category carrying an action of a free partially commutative monoid. We calculate this dimension in the case that the abelian category has infinite coproducts and enough projectives. Previously the author solved the same problem for abelian categories with exact coproducts.     

The Homology of Partial Monoid Actions and Petri Nets

The aim of this paper is to study the homology theory of partial monoid actions and apply it to computing the homology groups of mathematical models for concurrency. We study the Baues–Wirsching homology groups of a small category associated with a partial monoid action on a set. We prove that these groups can be reduced to the Leech homology groups of the monoid. For a trace monoid with a partial action on a set, we build a complex of free Abelian groups for computing the homology groups of this small category. It allows us to solve the problem posed by the author on the construction of an algorithm to computing the homology groups of elementary Petri nets. We describe the algorithm and give examples of computing the homology groups.     

The Structure of Normal Algebraic Monoids

We show that any normal algebraic monoid is an extension of an abelian variety by a normal affine algebraic monoid. This extends (and builds on) Chevalley's structure theorem for algebraic groups.     

Some results about zero‐cycles on abelian and semi‐abelian varieties

In this short note we extend some results obtained in [7]. First, we prove that for an abelian variety A with good ordinary reduction over a finite extension of with p an odd prime, the Albanese kernel of A is the direct sum of its maximal divisible subgroup and a torsion group. Second, for a semi‐abelian variety G over a perfect field k, we construct a decreasing integral filtration of Suslin's singular homology group, , such that the successive quotients are isomorphic to a certain Somekawa K‐group.     

Improved homological stability for the mapping class group with integral or twisted coefficients

In this paper we prove stability results for the homology of the mapping class group of a surface. We get a stability range that is near optimal, and extend the result to twisted coefficients.     

Künneth decompositions for quotient varieties

In this paper we discuss Künneth decompositions for finite quotients of several classes of smooth projective varieties. The main result is the existence of an explicit (and readily computable) Chow-Künneth decomposition in the sense of Murre with several pleasant properties for finite quotients of abelian varieties. This applies in particular to symmetric products of abelian varieties and also to certain smooth quotients in positive characteristics which are known to be not abelian varieties, examples of which were considered by Enriques and Igusa. We also consider briefly a strong Künneth decomposition for finite quotients of projective smooth linear varieties.     

Orbit decompositions of orthogonal monoids and the orders of finite orthogonal monoids

In this paper we determine the \(G\times G\) orbits of both an even orthogonal monoid and an even special orthogonal monoid, where G is the unit group of the even special orthogonal monoid. We then use the orbit decompositions to compute the orders of these monoids over a finite field.     

Cohomology monoids of monoids with coefficients in semimodules II

We relate the old and new cohomology monoids of an arbitrary monoid M with coefficients in semimodules over M, introduced in the author’s previous papers, to monoid and group extensions. More precisely, the old and new second cohomology monoids describe Schreier extensions of semimodules by monoids, and the new third cohomology monoid is related to a certain group extension problem.     

On certain quotients of the Green rings of dihedral 2-groups

In this paper we study the properties of Green rings of dihedral 2-groups, and in particular certain quotients of these Green rings introduced by Benson and Carlson. It is shown that these quotients can be realised as group rings over . The properties of the corresponding groups are investigated: they are shown to be abelian, torsion-free and infinitely generated. We also show how taking products of elements of these groups is related to the structure of the Auslander–Reiten quivers for dihedral 2-groups.     

The support of a uniform semigroup valued measure

In this paper we introduce the notion of orthogonality for arbitrary families of elements of a partial abelian monoid, we study uniform semigroup valued measures on a partial abelian monoid, and we establish interesting results about the existence of the support of such measures. Received July 16, 2005; accepted in final form December 13, 2005.