Jane Gilman

On the Nielsen Type and the Classification for the Mapping Class Group

Let S be a compact surface of genus g with n boundary components, with 2g - 2 + n > 0, and let M(S) be its mapping class group (also known as the Teichmiiller modular group). Using measurd foliations, Thurston [ 15 ] has shown that M(S) contains three different types of elements. Bers has shown this decomposition can be obtained using Teichmiiller distance [3]. Subdividing one of the classes further, he obtains four classes. Nielsen has a series of papers [7; 8, Parts I-III] in which he develops an elaborate theory of fixed points of mapping classes. If t is any homeomorphism of S and h a lift of t (or of a power of 1) to the upper half plane, Nielsen’s theory involves the assignment of a pair of integers to h, called the Nielsen type of h. The purpose of this paper is to show how the Thurston classes and the Bers classes can be defined in terms of the Nielsen types of the lifts of

Discreteness criteria and the hyperbolic geometry of palindromes

We consider non-elementary representations of two generator free groups in PSL(2;C), not necessarily discrete or free, G = hA;Bi. A word in A and B, W (A;B), is a palindrome if it reads the same forwards and backwards. A word in a free group is primitive if it is part of a minimal generating set. Primitive elements of the free group on two generators can be identied with the positive rational numbers. We study the geometry of palindromes and the action of G in H3 whether or not G is discrete. We show that there is a core geodesic L in the convex hull of the limit set of G and use it to prove three results: the rst is that there are well dened maps from the non- negative rationals and from the primitive elements to L; the second is that G is geometrically nite if and only if the axis of every non-parabolic palindromic word in G intersects L in a compact interval; the third is a description of the relation of the pleating locus of the convex hull boundary to the core geodesic and to palindromic elements.

KLEINIAN GROUPS WITH REAL PARAMETERS

We find all real points of the analytic space of two generator Mobius groups with one generator elliptic of order two. Geometrically this is a certain slice through the space of two generator discrete groups, analogous to the Riley slice, though of a very different nature. We obtain applications concerning the general structure of the space of all two generator Kleinian groups and various universal constraints for Fuchsian groups.

Algorithms, complexity and discreteness criteria in PSL(2, C)

Let G be a subgroup of PSL(2, C). The discreteness problem for G is the problem of determining whether or not G is discrete. In this paper we assume that all groups considered are non-elementary. If G is generated by two elements, A and B of PSL(2, C), we have what is called the two-generator discreteness problem. Two-generator groups are important because, by a result of Jorgensen, an arbitrary subgroup of PSL(2, C) is discrete if and only if every non-elementary two-generator subgroup is [8]. One solution to the two-generator real discreteness problem (i.e., A and B in PSL(2, R)) is a geometrically motivated algorithm which was begun in [7] and completed in [6], where the algorithm is given in three forms. Our goal here is to compute the computational complexity of the three forms of the PSL(2, R) discreteness algorithm that appear in [6] and of the algorithm restricted to PSL(2, Q). While this may seem far afield from the original mathematical problem of determining discreteness, it settles the question as to in what sense the discreteness problem requires an algorithm. That is, the computational complexity of the algorithm can be used to prove that an algorithmic approach is necessary and that the geometric algorithm of [6] is the best discreteness condition that one can hope to obtain. We also investigate the computational complexity of several other discreteness criteria, including Rileys PSL(2, C) procedure [15], which is not always an algorithm, and Jorgensens inequality [8]. We find that the geometric two-generator PSL(2, Q) algorithm is of linear complexity. By contrast, Rileys procedure appears to be at least exponential even when restricted to the two-generator PSL(2, Q) case; but it has never been completely analyzed.

Classical two-parabolicT-Schottky Groups

AT-Schottky group is a discrete group of Möbius transformations whose generators identify pairs of possibly-tangent Jordan curves on the complex sphere ℂ. If the curves are Euclidean circles, then the group is termed classicalT-Schottky.We describe the boundary of the space of classicalT-Schottky groups affording two parabolic generators within the larger parameter space of allT-Schottky groups with two parabolic generators. This boundary is surprisingly different from that of the larger space. It is analytic, while the boundary of the larger space appears to be fractal. Approaching the boundary of the smaller space does not correspond to pinching; circles necessarily become tangent, but extra parabolics need not develop.As an application, we construct an explicit one parameter family of two parabolic generator non-classicalT-Schottky groups.

A matrix representation for automorphisms of compact Riemann surfaces

In this paper we prove a result which has as corollaries theorems of Hurwitz, Accola, Grothendieck, and Serre on automorphisms of Riemann surfaces.

Determining Thurston classes using Nielsen types

In previous work [3] we showed how the Thurston or Bers classifications of diffeomorphisms of surfaces could be obtained using the Nielsen types of the lifts of the diffeomorphism to the unit disc. In this paper we find improved conditions on the Nielsen types for the Thurston and Bers classes. We use them to verify that an example studied by Nielsen is a pseudo-Anosov diffeomorphism with stretching factor of degree 4. This example is of interest in its own right, but it also serves to illustrate exactly how the Nielsen types are used for verifying examples. We discuss the general usefulness of this method.

A geometric approach to the hyperbolic Jørgensen inequality

where tr is the trace and [ , ] represents the commutator. This is one of the most useful and powerful tools available for determining nondiscreteness. The precise geometric meaning of this inequality has been unclear. Here we first give an equivalent but more geometric formulation of the inequality (see (II)) in the case of hyperbolic elements in PSL(2,R). Next we outline a proof of the inequality for the case of purely hyperbolic subgroups of PSL(2, R) which shows that under these circumstances an even stronger inequality is satisfied (see (III)). For purely hyperbolic discrete groups (I) is trivially satisfied for all but the finite number of conjugacy classes of elements (or their inverses) with multiplier between 1 and (3 4\/5)/2, whereas the stronger inequality is not. Groups of Möbius transformations in space are an increasingly important area of study. A J0rgensen type inequality would be significant there, and it is hoped that this formulation of J0rgensens inequality might point the way toward the proper formulation in space. I wish to thank Professor B. Maskit for pointing out to me the significance of this formulation, and Professor D. Gallo, who was the first to ask whether the inequalities in [G] implied J0rgensens inequality. The inequalities. Let A and B be hyperbolic matrices with multipliers R and K respectively. If VA, WA and vs , WB are the repelling and attracting fixed points of A and B respectively, let C be the cross ratio

Geometry of Riemann Surfaces: Cutting sequences and palindromes

We give a unified geometric approach to some theorems about primitive elements and palindromes in free groups of rank 2. The geometric treatment gives new proofs of the theorems. Dedicated to Bill Harvey on his 65th birthday.

On the existence of elliptics in subgroups of PSL (2, ℝ): A graphical picture

Let F bea non-elementary subgroup of PSL(2, ℝ). The purpose of this paper is to elucidate the nature and the extent of elliptic elements in F.