Hidden symmetries and commensurability of $2$-bridge link complements

Christian Millichap and William Worden

Vol. 285 (2016), No. 2, 453–484

DOI: 10.2140/pjm.2016.285.453

Abstract

In this paper, we show that any nonarithmetic hyperbolic $2$-bridge link complement admits no hidden symmetries. As a corollary, we conclude that a hyperbolic $2$-bridge link complement cannot irregularly cover a hyperbolic $3$-manifold. By combining this corollary with the work of Boileau and Weidmann, we obtain a characterization of $3$-manifolds with nontrivial JSJ-decomposition and rank-two fundamental groups. We also show that the only commensurable hyperbolic $2$-bridge link complements are the figure-eight knot complement and the $6_2^2$ link complement. Our work requires a careful analysis of the tilings of ${ℝ}^2$ that come from lifting the canonical triangulations of the cusps of hyperbolic $2$-bridge link complements.

Keywords

2-bridge links, hidden symmetries, commensurability

Mathematical Subject Classification 2010

Primary: 57M25, 57M50

Milestones

Received: 5 January 2016

Revised: 5 July 2016

Accepted: 6 July 2016

Published: 21 November 2016