Statistics of finite degree covers of torus knot complements. (Statistiques des revêtements de degré fini des compléments de nœuds toriques.) (English. French summary) Zbl 1535.20132
Summary: In the first part of this paper, we determine the asymptotic subgroup growth of the fundamental group of a torus knot complement. In the second part, we use this to study random finite degree covers of torus knot complements. We determine their Benjamini-Schramm limit and the linear growth rate of the Betti numbers of these covers. All these results generalise to a larger class of lattices in \(\mathrm{PLS}(2, \mathbb{R})\times\mathbb{R}\). As a by-product of our proofs, we obtain analogous limit theorems for high index random subgroups of non-uniform Fuchsian lattices with torsion.
MSC:
| 20E07 | Subgroup theorems; subgroup growth |
| 22E40 | Discrete subgroups of Lie groups |
| 57M50 | General geometric structures on low-dimensional manifolds |
| 20F34 | Fundamental groups and their automorphisms (group-theoretic aspects) |
| 57K10 | Knot theory |
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