Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Cones and ping-pong in three dimensions

Gabriel Frieden, Félix Gélinas, Étienne Soucy

Abstract

We study the hypergeometric group in ${\rm GL}_3(\mathbb{C})$ with parameters $α= (\frac{1}{4}, \frac{1}{2}, \frac{3}{4})$ and $β= (0,0,0)$. We give a new proof that this group is isomorphic to the free product $\mathbb{Z}/4\mathbb{Z} * \mathbb{Z}/2\mathbb{Z}$ by exhibiting a ping-pong table. Our table is determined by a simplicial cone in $\mathbb{R}^3$, and we prove that this is the unique simplicial cone (up to sign) for which our construction produces a valid ping-pong table.

Cones and ping-pong in three dimensions

Abstract

We study the hypergeometric group in with parameters and . We give a new proof that this group is isomorphic to the free product by exhibiting a ping-pong table. Our table is determined by a simplicial cone in , and we prove that this is the unique simplicial cone (up to sign) for which our construction produces a valid ping-pong table.
Paper Structure (12 sections, 6 theorems, 59 equations, 4 figures)

This paper contains 12 sections, 6 theorems, 59 equations, 4 figures.

Table of Contents

  1. Introduction
  2. A three-dimensional ping-pong table
  3. Cones
  4. Free products and the ping-pong lemma
  5. A ping-pong table in $\mathbb{R}^3$
  6. Matrix logarithms
  7. Uniqueness of the cone $C$
  8. Two-dimensional projection
  9. Definition of the projection
  10. Uniqueness revisited
  11. A smaller ping-pong table
  12. Comparison with the two-dimensional and four-dimensional cases

Key Result

Lemma 2.2

Let $G,H$ be two non-trivial subgroups of a group $K$, such that at least one of $G$ and $H$ has more than two elements. Suppose $K$ acts on a set $S$, and there are two non-empty subsets $X,Y \subset S$ satisfying the following properties: Then the subgroup of $K$ generated by $G$ and $H$ is a free product; that is, $\langle G, H \rangle = G*H$.

Figures (4)

  • Figure 1: The large triangle in the first quadrant (colored red) is the projection of $X = \pm C$. The three large triangles in the other quadrants are the projections of $RX, R^2X$, and $R^3X$. The smaller triangles in the first quadrant are the projections of $T(RX), T(R^2X)$, and $T(R^3X)$.
  • Figure 2: The points $(TR)^t(v)$ (left) and $(TR^{-1})^t(u)$ (right) in the 2D projection.
  • Figure 3: The cones $Q_1$ and $Q_2$.
  • Figure 4: The cones $X'_1$ and $X'_2$, whose intersection is $X'$.

Theorems & Definitions (15)

  • Conjecture 1.1
  • Remark 1.2
  • Definition 2.1
  • Lemma 2.2: LyndonSchupp
  • Theorem 2.3
  • proof
  • Theorem 3.1
  • Lemma 3.2
  • proof
  • Remark 3.3
  • ...and 5 more