Hyperbolic Integral Homology Spheres and Binary Icosahedral Representations

TL;DR

This paper explores the representations of hyperbolic integral homology spheres into the binary icosahedral group, revealing that such manifolds have quotient dimensions only 2 or 3, with infinitely many examples for each case.

Contribution

It provides a geometric interpretation of $2I$ representations and classifies hyperbolic 3-manifolds by their quotient dimension, showing the existence of infinitely many manifolds in each class.

Findings

Hyperbolic 3-manifolds have quotient dimension 2 or 3.

Manifolds with no non-trivial $A_5$ representations have quotient dimension 3.

Infinite families of manifolds with quotient dimension 2 are constructed via Dehn surgery.

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.