Roger C. Alperin

The boundary of the Gieseking tree in hyperbolic three-space

Abstract We give an elementary proof of the Cannon–Thurston Theorem in the case of the Gieseking manifold. We do not use Thurstons structure theory for Kleinian groups but simply calculate with two-by-two complex matrices. We work entirely on the boundary, using ends of trees, and obtain pictures of the regions which are successively filled in by the Peano curve of Cannon and Thurston.

The Modular Tree of Pythagoras

The Pythagorean triples that are relatively prime (called the primitive triples) have the elementary and beautiful characterization as integers x = m2 − n2, y = 2mn, z = m2 + n2 (when y is even) for relatively prime integers m and n of opposite parity. One can think of this as replacing the parameter t for the circle with the fraction m/n and then scaling. Our motivation for understanding the triples stems from the realization that one can enumerate the rational numbers on the line by using the modular group, in a sense reversing the Euclidean algorithm [2]. Now the line can be transformed by a linear fractional transformation to the circle. This transformation changes fractions to rational points on the circle, and after scaling this process gives rise to Pythagorean triples. Roughly speaking, we can establish a correspondence of a Pythagorean triple [m2 − n2, 2mn,m2 + n2] in which m and n are relatively prime with a matrix belonging to SL2(Z) (the group of two-by-two integral matrices of determinant one) whose entries depend on m and n. Since the modular group = PSL2(Z) = SL2(Z)/{±I } is essentially a free group, it follows that there is an underlying tree structure to Pythagorean triples. Making this tree structure and its connection to the modular group explicit is a bit delicate, but the payoff is worth the effort. Our main results can be summarized as follows:

A Strong Schottky Lemma for Nonpositively Curved Singular Spaces

Motivated by the question of faithfulness of the four-dimensional Burau representation, we study a generalization of the Schottky Lemma which is useful for studying actions on affine buildings.

Linear representations of binate groups

Binate groups are acyclic groups having no subgroups of finite index, and all finitely generated residually finite normal subgroups are central. Using homological arguments, we show that there are no non-trivial finite-dimensional unitary representations. With some results from algebraic groups, we can further show that there are no non-trivial finite-dimensional linear representations.

Uniform Growth of Polycyclic Groups

We prove that polycyclic groups have uniform exponential or polynomial growth.

Trisections and Totally Real Origami

1. CONSTRUCTIONS IN GEOMETRY. The study of methods that accomplish trisections is vast and extends back in time approximately 2300 years. My own favorite method of trisection from the Ancients is due to Archimedes, who performed a “neusis” between a circle and line. Basically a neusis (or use of a marked ruler) allows the marking of points on constructed objects of unit distance apart using a ruler placed so that it passes through some known (constructed) point P . Here is Archimedes’ trisection method (see Figure 1): Given an acute angle between rays r and s meeting at the point O, construct a circle K of radius one at O, and then extend r to produce a line that includes a diameter of K . The circle K meets the ray s at a point P . Now place a ruler through P with the unit distance CD lying with C on K and D on the ray opposite to r . That the angle ODP is the desired trisection is easy to check using the isosceles triangles DCO and COP and the exterior angle of the triangle PDO. As one sees when trying this for oneself, there is a bit of “fiddling” required to make everything line up as desired; that fiddling is also essential when one does origami.

Platonic triangles of groups

Research supported by NSA and NSF. 1980 Mathematics Subject Classi cation (1985 Revision): 20E99.

Integral Sets and the Center of a Finite Group

We give a description of the atoms in the Boolean algebra generated by the integral subsets of a finite group.

Undistorted solvable linear groups

We discuss distortion of solvable linear groups over a locally compact field and provide necessary and sufficient conditions for a subgroup to be undistorted when the field is of characteristic zero.

Free products of matrices

In this note we give some new examples of matrix groups which are free products. We use this in our study of the Burau representation in dimension four.