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Classification of parabolic generating pairs of Kleinian groups with two parabolic generators. (English) Zbl 1458.57024

The purpose of the paper under review is to reprove the following classification theorem announced by Agol and proved in [H. Akiyoshi et al., Trans. Am. Math. Soc. 374, No. 3, 1765–1814 (2021; Zbl 1458.57025)]:
Theorem.
(1) Every hyperbolic \(2\)-bridge link group has precisely two parabolic generating pairs up to equivalence.
(2) Every Heckoid group has a unique parabolic generating pair up to equivalence.
Using the above theorem combined with other theorems in some different references, the authors not only completely characterize epimorphisms between \(2\)-bridge knot groups but also completely characterize degree one maps between the exterior of hyperbolic \(2\)-bridge links.

MSC:

57M50 General geometric structures on low-dimensional manifolds
57K10 Knot theory
30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)

Citations:

Zbl 1458.57025
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References:

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