Vol. 285, No. 2, 2016

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ISSN: 0030-8730
Hidden symmetries and commensurability of $2$-bridge link complements

Christian Millichap and William Worden

Vol. 285 (2016), No. 2, 453–484

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 622 link complement. Our work requires a careful analysis of the tilings of 2 that come from lifting the canonical triangulations of the cusps of hyperbolic 2-bridge link complements.

2-bridge links, hidden symmetries, commensurability
Mathematical Subject Classification 2010
Primary: 57M25, 57M50
Received: 5 January 2016
Revised: 5 July 2016
Accepted: 6 July 2016
Published: 21 November 2016
Christian Millichap
Department of Mathematics
Linfield College
McMinnville, OR 97128
United States
William Worden
Department of Mathematics
Temple University
Philadelphia, PA 19122
United States