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Hyperbolic Integral Homology Spheres and Binary Icosahedral Representations

Maria Stuebner

Abstract

This paper examines the representations of hyperbolic integral homology spheres into the binary icosahedral group $2I$. We specifically give a geometric meaning to $2I$ representations by relating them to Larsen's notion of quotient dimension, which gives us a sense of the frequency of regular finite covers. Our main theorem shows that hyperbolic 3-manifolds can only have quotient dimension 2 or 3, and each case is obtained infinitely many times. More specifically, we show that those with no non-trivial $A_5$ representations have quotient dimension 3, and we find a family of hyperbolic 3-manifolds obtained by Dehn surgery on an infinite 2-bridge hyperbolic knot family with quotient dimension 2.

Hyperbolic Integral Homology Spheres and Binary Icosahedral Representations

Abstract

This paper examines the representations of hyperbolic integral homology spheres into the binary icosahedral group . We specifically give a geometric meaning to representations by relating them to Larsen's notion of quotient dimension, which gives us a sense of the frequency of regular finite covers. Our main theorem shows that hyperbolic 3-manifolds can only have quotient dimension 2 or 3, and each case is obtained infinitely many times. More specifically, we show that those with no non-trivial representations have quotient dimension 3, and we find a family of hyperbolic 3-manifolds obtained by Dehn surgery on an infinite 2-bridge hyperbolic knot family with quotient dimension 2.

Paper Structure

This paper contains 14 sections, 6 theorems, 59 equations, 5 figures, 2 tables.

Table of Contents

  1. Outline
  2. Quotient dimension.
  3. Hurwitz Surfaces
  4. Quotient dimension cannot be 1.
  5. Some Normal Linear Subgroups of $G$
  6. Quotient dimension can sometimes be 2.
  7. Constructing $G(\mathbb{C})$
  8. Semi-direct Product
  9. Lifting $A_5$ to the Binary Icosahedral Group $2I$
  10. Quaternions
  11. 2-Bridge Knots
  12. Quotient dimension can sometimes be 3.
  13. Open Questions
  14. Code

Key Result

Theorem 1

Let $Y$ be a hyperbolic 3-manifold, and let $\Gamma = \pi_1(Y)$. Then, the quotient dimension of $\Gamma$ is either 2 or 3, and each case is obtained infinitely many times. More specifically,

Figures (5)

  • Figure :
  • Figure :
  • Figure :
  • Figure :
  • Figure :

Theorems & Definitions (13)

  • Conjecture : Kirby, 3.15A
  • Definition 1: Larsen, 0.2
  • Theorem 1
  • Theorem 1.1: Larsen, 0.3
  • Proposition 1.2
  • proof
  • Definition 2.1: Larsen, 1.1
  • Lemma 2.1: Larsen, 1.2
  • Proposition 2.2: Larsen, 1.3
  • Proposition 2.3
  • ...and 3 more