Limit Sets of Kleinian Groups.

Limit Set of a Quasifuchsian Group Representing a Pair of Punctured Tori (in the Parameters of Troels Jorgensen)

Introduction

The pictures were created in a project conducted by David Mumford at Harvard, assisted primarily by Curt McMullen and myself. (Curt McMullen has made available many of his papers and pictures here.) The starting date of this project was 1980, at the same time Mandelbrot was lecturing at Harvard on Julia sets and what has now come to be called the Mandelbrot set.

Here is a collection of topics to explore

Limit Sets of Schottky Groups

Schottky groups have the simplest geometry among kleinian groups, and the easiest limit sets to think about and compute.

Quasifuchsian Groups and Further Deformation

Quasifuchsian groups may be considered as limits of Schottky groups where the dust of the limit set coagulates into continuous curves. These curves can then become a lacy network with further deformation.

Limit Sets of Punctured Surface Groups

Here we present pictures of terminal regular b-groups representing arbitrary punctured surfaces of finite type. The program requires knowledge of a side-pairing set of generators. These are even more elaborate laceworks than the quasifuchsian groups.

Limit Sets of Degenerate Groups

Here we present pictures of degenerate limit sets, which are either sphere-filling curves or have Hausdorff dimension 2. (And so they are pretty hard to draw!)

Fortran Program kleinian

This is a discussion of the Fortran programs we have written to draw these images, which are all computed originally as PostScript. It runs under Gnu FORTRAN 77 and Linux.

Limit Sets of Schottky Groups

	Compact Riemann surfaces of genus g can be constructed by a very general procedure. For genus 2, we take two linear fractional transformations T1 and T2 that satisfy a special geometric condition. There should be four simple closed curves C1, C2, C3, and C4, with mutually disjoint interiors such that T1(Interior of C3) = (Exterior of C1) and T2(Interior of C4) = (Exterior of C2) The first picture shows a pattern of circles with these conditions fulfilled. (Click to see enlarged version.)
	The second picture shows the images of the basic circles under the action of all words of length at most 3 in the generators and their inverses. (Click to enlarge.)
	The third picture shows the limit set of the given Schottky group which consists of the accumulation points of the infinitely many nested circles obtained from the basic circles by applying the elements of the group. It is just a barely visible scattering of fractal dust. (Click to enlarge.)

Quasifuchsian Limit Set

Here's a PostScript version of this picture:

double-spiral.epsi.gz

A quasifuchsian group is a quasiconformal deformation of a Fuchsian group. A Fuchsian group is a discrete subgroup of PSL₂(R) or more generally any discrete subgroup of PSL₂(C) that possesses an invariant circle in the Riemann sphere.

To say G₂ is a quasiconformal deformation of G₁ means that there is a quasiconformal homeomorphism w of the Riemann sphere onto itself such that

G₂ = w^-1 G₁ w

The chief significance lies in the subject of uniformization. Koebe's uniformization says that any Riemann surface S of negative Euler characteristic may be realized as H/G for some Fuchsian group G. The lower halfplane then represents the "mirror" image of S. Possibly Lipman Bers' most elegant theorem (there are many choices) is that of Simultaneous Uniformization: for any pair of homeomorphic Riemann surfaces S₁, S₂ of negative Euler characteristic and finite topological type (i.e. finite genus and finitely many special points) there is a quasifuchsian group G which represents both S₁ and S₂ in its two invariant domains (the two components of the ordinary set). One main area of research in uniformization is to what extent this can be generalized to collections of Riemann surfaces of various kinds.

The Problem of Necks

This picture is an enlargement of a portion of the previous limit set. The two points marked with arrows are limit points corresponding to two infinite words in the free group which are extremely close in the lexicographical ordering on the free group. Yet, nonetheless there is an extremely large and visible portion of the limit set stretching between these two points. This is the problem of "necks."

Double Cusp

This is a picture of the limit set of a free group on two generators in which two words have become "accidentally parabolic". It is a limiting case of a quasifuchsian group representing a pair of punctured tori. If the generators are denoted X and Y, then the words that are parabolic are Y and X¹⁵ Y. This is called the 1/15 double cusp.