Single Boundary Component Research Articles

Horoballs and iteration of holomorphic maps on bounded symmetric domains

Given a fixed-point free compact holomorphic self-map f on a bounded symmetric domain D, which may be infinite dimensional, we establish the existence of a family {H(ξ,λ)}λ>0 of convex f-invariant domains at a point ξ in the boundary ∂D of D, which generalises completely Wolff's theorem for the open unit disc in C. Further, we construct horoballs at ξ and show that they are exactly the f-invariant domains when D is of finite rank. Consequently, we show in the latter case that the limit functions of the iterates (fn) with weakly closed range all accumulate in one single boundary component of ∂D.

Erratum to: Pairs of boundary slopes with small differences

1. The second boundary slope in Theorem 1 is described inappropriately as the fraction seems to be reducible by 2. The value of the fraction is correct, and so Theorem 1 is completely true. 2. Corollary 2 was not true for the second surface. It should be corrected as: Let F be the essential surface constructed in the proof of Theorem 1. Then F has a single boundary component, and must be non-orientable. 3. Proposition 2 was also not true for the second surface. It should be corrected as: For essential surfaces F and F ′ constructed in the proof of Theorem 1, the values −χ s are both equal to 1. This implies that their Euler characteristics are −n and −n + 7, respectively. In particular, the Euler characteristic of F is equal to the denominators of their boundary slopes. 4. The comments just after Proposition2were not correct aswasProposition2. In fact, paper [1] has been updated recently, and it is now including extensions to Theorem 1, Corollary 2, and Proposition 2. Please refer [1] for detailed explanations.

Generalized north–south dynamics on the space of geodesic currents

We prove uniform north-south dynamics type results for the action of $\varphi\in Out(F_{N})$ on the space of projectivized geodesic currents $\mathbb{P}Curr(S)=\mathbb{P}Curr(F_{N})$, where $\varphi$ is induced by a pseudo-Anosov homeomorphism on a compact surface S with boundary such that $\pi_{1}(S)=F_{N}$. As an application, we show that for a subgroup $H\le Out(F_N)$, containing an iwip, either $H$ contains a hyperbolic iwip or $H$ is contained in the image in $Out(F_N)$ of the mapping class group of a surface with a single boundary component.

CLOSED ESSENTIAL SURFACES AND WEAKLY REDUCIBLE HEEGAARD SPLITTINGS IN MANIFOLDS WITH BOUNDARY

We show that there are infinitely many two component links in S3 whose complements have weakly reducible and irreducible non-minimal genus Heegaard splittings, yet the construction given in the theorem of Casson and Gordon does not produce an essential closed surface. The situation for manifolds with a single boundary component is still unresolved though we obtain partial results regarding manifolds with a non-minimal genus weakly reducible and irreducible Heegaard splitting.

Complexity and Heegaard genus of an infinite class of compact 3-manifolds

Using the theory of hyperbolic manifolds with totally geodesic boundary, we provide for every n ≥ 2 a class M n of such manifolds all having Matveev complexity equal to n and Heegaard genus equal to n + 1. All the elements of M n have a single boundary component of genus n, and #M n grows at least exponentially with n.

Local connectivity, Kleinian groups and geodesics on the blowup of the torus

Let N = H 3 / be a hyperbolic 3-manifold with free fundamental group π1(N) ∼ ∼ h A, Bi , such that (A, B) is parabolic. We show that the limit setof N is always locally connected. More precisely, letbe a compact surface of genus 1 with a single boundary component, equipped with the Fuchsian action of π1(�) on the circle S 1 1 . We show that for any homotopy equivalence f : � → N, there is a natural continuous map

Taut foliations in punctured surface bundles, I

Given a fibred, compact, orientable 3-manifold with single boundary component, we show that a fibration with fiber surface of negative Euler characteristic can be perturbed to yield taut foliations realizing an open interval of boundary slopes about the boundary slope of the fibration. These taut foliations extend to taut foliations in the corresponding surgery manifolds. 2000 Mathematics Subject Classification: primary 57M25; secondary 57R30.

A characterization of knot polynomials

IN [7] Seifert presents a method for constructing a knot in the three-sphere S3, by imbedding a surface (with a single boundary component) in S” with prescribed self-linking characteristics. This technique was used to characterize the Alexander polynomial of a knot, in an algebraic manner. But the Alexander polynomial is only the first of a sequence of polynomials which arise as knot invariants, and it is natural to consider the problem of characterizing them algebraically. Seifert’s construction does not seem delicate enough to carry out this task.f It is the aim of this note to present a new technique for constructing knots, also based on a prescription of linking information, which can resolve the problem. The techniques to be presented here generalize to give similar characterizations of suitably defined invariants of higher-dimensional knots. This entire subject will be dealt with in a future paper; also see [4, Part II].