Given the fundamental group 0 of a finite-volume complete hyperbolic 3-manifold M, it is possible to associate to any representation ρ : 0 → Isom(H3) a numerical invariant called volume. This invariant is bounded by the hyperbolic volume of M and satisfies a rigidity condition: If the volume of ρ is maximal, then ρ must be conjugated to the holonomy of the hyperbolic structure of M. This paper generalizes this rigidity result by showing that if a sequence of representations of 0 into Isom(H3) satisfies limn→1Vol(ρn) = Vol(M), then there must exist a sequence of elements gn 2 Isom(H3) such that the representations gn o ρn o g-1 n converge to the holonomy of M. In particular if the sequence (ρn)n2N converges to an ideal point of the character variety, then the sequence of volumes must stay away from the maximum. In this way we give an answer to [16, Conjecture 1]. We conclude by generalizing the result to the case of k-manifolds and representations in Isom(Hm), where m ≥ k ≥ 3.
Francaviglia, S., Savini, A. (2020). Volume rigidity at ideal points of the character variety of hyperbolic 3-manifolds. ANNALI DELLA SCUOLA NORMALE SUPERIORE DI PISA. CLASSE DI SCIENZE, 2020(4), 1325-1344 [10.2422/2036-2145.201709_010].
Volume rigidity at ideal points of the character variety of hyperbolic 3-manifolds
Savini, A
2020
Abstract
Given the fundamental group 0 of a finite-volume complete hyperbolic 3-manifold M, it is possible to associate to any representation ρ : 0 → Isom(H3) a numerical invariant called volume. This invariant is bounded by the hyperbolic volume of M and satisfies a rigidity condition: If the volume of ρ is maximal, then ρ must be conjugated to the holonomy of the hyperbolic structure of M. This paper generalizes this rigidity result by showing that if a sequence of representations of 0 into Isom(H3) satisfies limn→1Vol(ρn) = Vol(M), then there must exist a sequence of elements gn 2 Isom(H3) such that the representations gn o ρn o g-1 n converge to the holonomy of M. In particular if the sequence (ρn)n2N converges to an ideal point of the character variety, then the sequence of volumes must stay away from the maximum. In this way we give an answer to [16, Conjecture 1]. We conclude by generalizing the result to the case of k-manifolds and representations in Isom(Hm), where m ≥ k ≥ 3.File | Dimensione | Formato | |
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