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**Hidden symmetries of hyperbolic links.**
*(English)*
Zbl 1503.57005

A hidden symmetry of a hyperbolic manifold \(M = \mathbf{H}^3/\Gamma\) is an automorphism (isometry) of a finite degree cover of \(M\) that does not descend to an automorphism of \(M\); such a hidden symmetry exists iff the commensurator \(\mathrm{Comm}(\Gamma)\) of \(\Gamma\) is strictly larger than its normalizer \(N(\Gamma)\) (in the isometry group of hyperbolic 3-space \(\mathbf{H}^3\)).

Neumann and Reid conjectured that the figure-eight knot and the two dodecahedral knots of B. Apanasov (ed.) et al. [Topology ’90. Contributions from a research semester in low dimensional topology, held at Ohio State University, Columbus, OH (USA), from Feb. through June 1990. Berlin: Walter de Gruyter (1992; Zbl 0747.00024)] are the only hyperbolic knots admitting hidden symmetries (i.e., their complements admit hidden symmetries). In the present paper, for any \(n \ge 4\) the author constructs non-arithmetic, \(n\)-component hyperbolic links whose complements \(\mathbf{H}^3/\Gamma_n\) admit hidden symmetries (by surgery on a chain link with \(n+1\) trivial components; note that an arithmetic link has always hidden symmetries since its normalizer \(N(\Gamma)\) is not discrete, and that the figure-8 knot is the only arithmetic hyperbolic knot); more precisely, the author poves that \(|\mathrm{Comm}(\Gamma_n) : N(\Gamma_n)| = n + 1\). Some examples of non-arithmetic links with hidden symmetries were known before, and in the introduction the author gives a short survey on knots and links without and with hidden symmetries.

Neumann and Reid conjectured that the figure-eight knot and the two dodecahedral knots of B. Apanasov (ed.) et al. [Topology ’90. Contributions from a research semester in low dimensional topology, held at Ohio State University, Columbus, OH (USA), from Feb. through June 1990. Berlin: Walter de Gruyter (1992; Zbl 0747.00024)] are the only hyperbolic knots admitting hidden symmetries (i.e., their complements admit hidden symmetries). In the present paper, for any \(n \ge 4\) the author constructs non-arithmetic, \(n\)-component hyperbolic links whose complements \(\mathbf{H}^3/\Gamma_n\) admit hidden symmetries (by surgery on a chain link with \(n+1\) trivial components; note that an arithmetic link has always hidden symmetries since its normalizer \(N(\Gamma)\) is not discrete, and that the figure-8 knot is the only arithmetic hyperbolic knot); more precisely, the author poves that \(|\mathrm{Comm}(\Gamma_n) : N(\Gamma_n)| = n + 1\). Some examples of non-arithmetic links with hidden symmetries were known before, and in the introduction the author gives a short survey on knots and links without and with hidden symmetries.

Reviewer: Bruno Zimmermann (Trieste)

### MSC:

57K10 | Knot theory |

57K32 | Hyperbolic 3-manifolds |

57M60 | Group actions on manifolds and cell complexes in low dimensions |

### Citations:

Zbl 0747.00024### References:

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[8] | M. Sakuma and J. Weeks, Examples of canonical decompositions of hyperbolic link complements, Japan. J. Math. (N.S.) 21 (1995), no. 2, 393-439. · Zbl 0858.57021 |

[9] | W. P. Thurston, The Geometry and Topology of Three-manifolds, Lecture notes, Princeton University Press, Princeton, 1979. |

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