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Cusp transitivity in hyperbolic 3-manifolds. (English) Zbl 1476.57025

This paper is interested in cusped, finite volume hyperbolic 3-manifolds \(M\) whose isometry group induces a \(k\)-transitive action on the set of cusps. A \(k\)-transitive action on the cusps is one where, for any two ordered \(k\)-tuples \((x_1, \ldots, x_k)\) and \((y_1, \ldots, y_k)\) of distinct cusps, there is an isometry \(g\) of \(M\) so that \(g(x_i) = y_i\) for \(1 \leq i \leq k\).
This paper follows one of R. Vogeler [Topol. Proc. 57, 25–36 (2021; Zbl 1445.57010)], who gave examples of \(M\) for \(k=1,2,3,4\) and showed that there is no bound on the number of cusps when \(M\) is 2-transitive. Specifically, this paper shows that \(k \leq 5\), and that in the event that \(k=5\) (respectively \(k=4\)), \(M\) must have 5 (respectively 4) cusps. The authors also give restrictions on \(M\) in the event that \(k=3\), including that if the action is \(3\)-transitive and not \(4\)-transitive, then \(M\) has 3, 5, 6 or 8 cusps.
The proofs of these results make use of the group theory of finite groups and permutation groups. For example, the result for \(k=5\) comes from fixing one cusp and considering the subgroup which acts \((k-1)\)-transitively on the other cusps. The properties of this group imply that \(k-1\) is at most 4.
The paper concludes by giving examples, as links in \(S^3\), of many of the cases which were shown to be possible. The paper does not give exhaustive lists, and leaves open the question of how many examples exist in various cases.

MSC:

57K32 Hyperbolic 3-manifolds
57M50 General geometric structures on low-dimensional manifolds
57M60 Group actions on manifolds and cell complexes in low dimensions

Citations:

Zbl 1445.57010

Software:

SnapPy
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Full Text: DOI arXiv

References:

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