Uniform Exponential Growth of Polycyclic Groups

We prove that polycyclic groups are of polynomial growth or of uniform exponential growth

Metabelian Wreath Products Are LERF

We show that a wreath product of two finitely generated abelian groups is LERF. Consequently the free metabelian groups are LERF.Comment: 3 page

Trisections and Totally Real Origami

We introduce a trisection axiom for mathematical origami and descibe the totally real origami numbers. We also discuss the solution of Alhazen's problem and its relation to trisections.Comment: 4 figure

Remarks on a Problem of Eisenstein

The fundamental unit of $\Z[\sqrt{N}]$ for square-free $N=5 mod 8$ is either $\epsilon$ or $\epsilon^3$ where $\epsilon$ denotes the fundamental unit of the maximal order of \Q(\sqrt{N}). We give infinitely many examples for each case.Comment: 4 page

The Modular Tree of Pythagorus

The Pythagorean triples have the structure of a ternary rooted tree; the tree is based on the Cayley graph of a free subgroup of the modular grou

Nonvanishing of algebraic entropy for geometrically finite groups of isometries of Hadamard manifolds

We prove that any geometrically finite (nonelementary) group of isometries of a pinched Hadamard manifold has uniform exponential growth.Comment: 6 figure

Uniform Growth, Actions on Trees and GL2GL_2

We use actions on trees to determine uniform exponential growth for subgroups of $GL_2$

A strong Schottky Lemma for nonpositively curved singular spaces

In this paper we give a criterion for pairs of isometries of a nonpositively curved metric space to generate a free group. This criterion holds only in singular spaces, for example in Euclidean buildings. The original motivation for our criterion was to prove that the four dimensional Burau representation is faithful. Although we do not settle this question, we do exhibit a related 2-parameter family of faithful representations of the free group F_2

2-Colorings of Cube Edges With 6 Each

This study was motivated by a problem posed by C. Morrow in her edge-colored cube constructions by origami. 1. Polya Counting The enumerator for all 2-colorings the edges of the cube where we use b of color B and w of color W is the coefficient of B b W w i

Solvable Groups of Exponential Growth and HNN Extensions

r which conjugates H 1 to H 2 ,#=<B,t|tH 1 t -1 = H 2 >. For solvable groups, a good example, is the group # 1 =<a,t|tat -1 = t 2 >. Many one relator groups have HNN decompositions; for example, consider # 2 =<a,t|a=[tat -1 ,t 2 at -2 ] >.This is, in fact, the HNN extension with base H =<a 0 ,a 1 ,a 2 | a 0 =[a 1 ,a 2 ] > and free subgroups F 1 =<a 0 = a, a 1 = tat -1 >,<F1