On a Theorem of Coxeter and Moser

November, 2012

The discussion on this web page falls under the topic of "geometric group theory". We want to look at symmetry groups of some familiar geometric objects. But at face value, the theorem we consider is one about group presentations.

Theorem (H.S.M. Coxeter and L. Moser, 1957):

Let G = < x,y,z | x^r = y^s = z^t = xyz=e >.
Then G is finite if and only if 1/r + 1/s + 1/t > 1

Here is an introduction to research in geometric group theory, with plenty of cool links, such as one to the KnotPlot page.

Group Presentations

In order to understand this beautiful theorem, we first need to know what a group presentation is. Examples are, up to isomorphism,
Z₈ = < x | x⁸ = e >
D₄ = < x,y | x⁴= y² =yxyx = e >
The free group on n letters is written
F_n = < x₁, x₂,..., x_n | e >
and contains all words in x₁, x₂,..., x_n x₁^-1, x₂^-1,..., x_n^-1 with no x_i, x_i^-1 adjacent to one another.

In general, a group presentation
G = < x₁, x₂...,, x_n | w₁ = w₂ = ..., = w_m = e >
is described by giving (for today) a finite number of generators x_i and a finite list of relations which are ``words'' w_j in the alphabet of all x_i and their inverses.

The Word Problem (M. Dehn, 1911)
Given a group presentation G and a word w in its generators and their inverses, is there an algorithm to decide if w=e?

Answer: No. In general, the word problem is undecidable. Here is a more detailed description of the word problem.

Tesselations of the Plane

Piles of Tiles at MIT's Media Lab.

Sorry! Dead link! Penrose tilings, inspired by Kepler

Sorry! Dead link! Kites and Darts

Here is an example of a Math Department using Penrose tilings for its floor design.

Introduction to tessellations

A bit of Escher

Seventeen plane symmetry groups, classified by Polya, also of interest to chemists.

Geometric viewpoint of the theorem of Coxeter and Moser.

Question: So what is the list of examples in the theorem?