Boundary slopes of non-orientable Seifert surfaces for knots

We study non-orientable Seifert surfaces for knots in the 3-sphere, and examine their boundary slopes. In particular, it is shown that for a crosscap number two knot, there are at most two slopes which can be the boundary slope of its minimal genus non-orientable Seifert surface, and an infinite family of knots with two such slopes will be described. Also, we discuss the existence of essential non-orientable Seifert surfaces for knots.     

Conjugacy separability of 1-acylindrical graphs of free groups

We use the theory of group actions on profinite trees to prove that the fundamental group of a finite, 1-acylindrical graph of free groups with finitely generated edge groups is conjugacy separable. This has several applications: we prove that positive, C′(1/6) one-relator groups are conjugacy separable; we provide a conjugacy separable version of the Rips construction; we use this latter to provide an example of two finitely presented, residually finite groups that have isomorphic profinite completions, such that one is conjugacy separable and the other does not even have solvable conjugacy problem.     

Divergence in 3-manifold groups

The divergence of the fundamental group of compact irreducible 3-manifolds satisfying Thurston's geometrization conjecture is calculated. For every closed Haken 3-manifold group, the divergence is either linear, quadratic or exponential, where quadratic divergence occurs precisely for graph manifolds and exponential divergence occurs when a geometric piece has hyperbolic geometry. An example is given of a closed 3-manifoldN with a Riemannian metric of nonpositive curvature such that the divergence is quadratic and such that there are two geodesic rays in the universal coverN whose divergence is precisely quadratic, settling in the negative a question of Gromov's.Partially supported by NSF grant DMS-9200433.     

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.     

Tits alternative for 3-manifold groups

Let M be an irreducible, orientable, closed 3-manifold with fundamental group G. We show that if the pro-p completion of G is infinite then G is either soluble-by-finite or contains a free subgroup of rank 2. Both authors are partially supported by “Bolsa de produtividade de pesquisa” from CNPq, Brazil. Received: 16 February 2006     

Simple quotients of hyperbolic 3-manifold groups

We show that hyperbolic -manifolds have residually simple fundamental group.

     

Braid groups of non-orientable surfaces and the Fadell-Neuwirth short exact sequence

Let M be a compact, connected non-orientable surface without boundary and of genus g?3. We investigate the pure braid groups P_n(M) of M, and in particular the possible splitting of the Fadell-Neuwirth short exact sequence

     

Immersions of non-orientable surfaces

Let F be a closed non-orientable surface. We classify all finite order invariants of immersions of F into R³, with values in any Abelian group. We show they are all functions of a universal order 1 invariant which we construct as T⊕P⊕Q, where T is a Z valued invariant reflecting the number of triple points of the immersion, and P,Q are Z/2 valued invariants characterized by the property that for any regularly homotopic immersions , P(i)−P(j)∈Z/2 (respectively, Q(i)−Q(j)∈Z/2) is the number mod 2 of tangency points (respectively, quadruple points) occurring in any generic regular homotopy between i and j.For immersion and diffeomorphism such that i and i○h are regularly homotopic we show:

P(i○h)−P(i)=Q(i○h)−Q(i)=(rank(h_∗−Id)+ε(deth∗∗))mod2     

Epimorphism sequences between hyperbolic 3-manifold groups

We will show that any infinite sequence of epimorphisms between finitely generated hyperbolic 3-manifold groups eventually consists of isomorphisms.

     

Every orientable Seifert 3-manifold is a real component of a uniruled algebraic variety

We show that any orientable Seifert 3-manifold is diffeomorphic to a connected component of the set of real points of a uniruled real algebraic variety, and prove a conjecture of János Kollár.     

Combings of semidirect products and 3-manifold groups

IfG is a finitely generated group that is abelian or word-hyperbolic andH is an asynchronously combable group then every split extension ofG byH is asynchronously combable. The fundamental group of any compact 3-manifold that satisfies the geometrization conjecture is asynchronously combable. Every split extension of a word-hyperbolic group by an asynchronously automatic group is asynchronously automatic.     

A sharper tits alternative for 3-manifold groups

The following theorem is proven. LetM be a closed, orientable, irreducible 3-manifold such that rankH ₁(M, ℤ/pℤ)≥3 for some primep. Then either π₁(M) is virtually solvable or it contains a free group of rank 2.     

On topologically stable singular surfaces in a 3-manifold

A formula for the Euler number of a generic singular surface in a 3-manifold is given. We also give an estimate of the number of connected components of the complement of such a surface.     

Ergodic actions of mapping class groups on moduli spaces of representations of non-orientable surfaces

Let M be a non-orientable surface with Euler characteristic χ(M) ≤ −2. We consider the moduli space of flat SU(2)-connections, or equivalently the space of conjugacy classes of representations

Minimal Euler characteristics of 4-manifolds with 3-manifold groups

Let π = π₁(M) for a compact 3-manifold M, and χ₄, p and q*be the invariants of Hausmann and Weinberger(1985), Kotschick(1994) and Hillman(2002), respectively. For a certain class of compact 3-manifolds M, including all those not containing two-sided RP2’s, we determine χ₄(π). We address when p(π)equals χ₄(π) and when q^*(π) equals χ₄(π), and answer a question raised by Hillman(2002).     

Conjugacy problem in groups of oriented geometrizable 3-manifolds

The aim of this paper is to show that the fundamental group of an oriented 3-manifold which satisfies Thurston's geometrization conjecture has a solvable conjugacy problem. In other words, for any such 3-manifold M, there exists an algorithm which can decide for any couple of elements u,v of π₁(M) whether u and v are in the same conjugacy class of π₁(M) or not. More topologically, the algorithm decides for any two loops in M, whether they are freely homotopic or not.     

Lengths of geodesics on non-orientable hyperbolic surfaces

We give an identity involving sums of functions of lengths of simple closed geodesics, known as a McShane identity, on any non-orientable hyperbolic surface with boundary which generalises Mirzakhani’s identities on orientable hyperbolic surfaces with boundary.