#### Vol. 285, No. 2, 2016

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

### Christian Millichap and William Worden

Vol. 285 (2016), No. 2, 453–484
##### 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.

##### Mathematical Subject Classification 2010
Primary: 57M25, 57M50
##### Milestones
Revised: 5 July 2016
Accepted: 6 July 2016
Published: 21 November 2016
##### Authors
 Christian Millichap Department of Mathematics Linfield College McMinnville, OR 97128 United States William Worden Department of Mathematics Temple University Philadelphia, PA 19122 United States