Pseudo-Anosov Mapping Classes and Their Representations by Products of Two Dehn Twists

Let S be a Riemann surface of analytically finite type （p, n） with 3p-3＋n 〉 0. Let a ∈ S and S = S - {a}. In this article, the author studies those pseudo-Anosov maps on S that are isotopic to the identity on S and can be represented by products of Dehn twists. It is also proved that for any pseudo-Anosov map f of S isotopic to the identity on S, there are infinitely many pseudo-Anosov maps F on S - {b} = S - {a, b}, where b is a point on S, such that F is isotopic to f on S as b is filled in.

Dehn Twists and Products of Mapping Classes of Riemann Surfaces with One Puncture

Let S be a Riemann surface that contains one puncture x. Let be the collection of simple closed geodesics on S, and let denote the set of mapping classes on S isotopic to the identity on S U {x}. Denote by tc the positive Dehn twist about a curve c ∈ . In this paper, the author studies the products of forms （tb^-m o t^na） o f^k, where a, b ∈ and f ∈ . It is easy to see that if a = b or a, b are boundary components of an x-punctured cylinder on S, then one may find an element f ∈ such that the sequence （tb^-m o t^na） ofk contains infinitely many powers of Dehn twists. The author shows that the converse statement remains true, that is, if the sequence （tb^-m o t^na） o f^k contains infinitely many powers of Dehn twists, then a, b must be the boundary components of an x-punctured cylinder on S and f is a power of the spin map tb^-1 o ta.

Fractional Dehn twists and modular invariants

In this paper, we establish a relationship between fractional Dehn twist coefficients of Riemann surface automorphisms and modular invariants of holomorphic families of algebraic curves. Specially, we give a characterization of pseudo-periodic maps with nontrivial fractional Dehn twist coefficients. We also obtain some uniform lower bounds of non-zero fractional Dehn twist coefficients.

Equivalent Separating Curves on Closed Surface of Genus 2

Let {A, B} be a complete system of the closed orientable surface F of genus 2. A simple closed curve C on F is separating with respect to （A, B） if it is disjoint from A U B and it cuts F into two once-punctured tori X, Y with A belong X, B belong Y. Let γ be a simple closed curve on F which is disjoint from A ∪ B and intersects C essentially in two points. In this paper, we show that up to isotopy, {hγ^n（C） ： n ∈ Z} is the set containing all the simple closed curves on F which is separating with respect to （A, B）, where hγ is the Dehn twist along γ on F. This also shows how two simple closed curves on F which are separating with respect to （A, B） are related. The result can be applied to yield all Haken spheres of a Heegaard splitting V∪F W which are weakly equivalent to a given Heken sphere of the splitting.

Remarks on Thurston's Construction of Pseudo-Anosov Maps of Riemann Surfaces

It is well known that certain isotopy classes of oseudo-Anosov maos on a Riemann surface S of non-excluded type can be defined through Dehn twists tα and tβ along simple closed geodesics α and β on S,respectively. Let G be the corresponding Fuchsian group acting on the hyperbolic plane H so that H/G≌S.For any point α∈S,define S = S／{α}.In this article, the author gives explicit parabolic elements of G from which he constructs pseudo-Anosov classes on S that can be projected to a given pseudo-Anosov class on S obtained from Thurston＇s construction.

Convex hull of set in thick part of Teichmller space

Let X be a non-elementary Riemann surface of type(g,n),where g is the number of genus and n is the number of punctures with 3g-3+n>1.Let T(X)be the Teichmller space of X.By constructing a certain subset E of T(X),we show that the convex hull of E with respect to the Teichmller metric,the Carathodory metric and the Weil-Petersson metric is not in any thick part of the Teichmler space,respectively.This implies that convex hulls of thick part of Teichmller space with respect to these metrics are not always in thick part of Teichmller space,as well as the facts that thick part of Teichmller space is not always convex with respect to these metrics.

On Reduced Lantern Relations in Mapping Class Groups

Let S be a hyperbolic Riemann surface with a finite area. Let G be the covering group of S acting on the hyperbolic plane H. In this paper, the author studies some algebraic relations in the mapping class group of S for S = S／{a point}. The author shows that the only possible relations between products of two Dehn twists and products of mapping classes determined by two parabolic elements of G are the reduced lantern relations. As a consequence, a partial solution to a problem posed by J. D. McCarthy is obtained.

A CLASS OF INCOMPRESSIBLE SURFACES IN 3-MANIFOLDS FIBERING OVER S^1

An important class in the study of topology of 3-manifolds is the so-called 'sufficiently large 3-manifolds', i.e. those compact 3-manifolds which contain properly embedded, 2-sided incompressible surfaces. We know that every sufficiently large 3-manifold can bc cut into some 3-cells along incompressible surfaces in a finite process. For a 3-manifold, it is interesting to know whether there exists an incompressible surface embedded in it and how the incompressible surface embeds.