Volume rigidity ad ideal points of the character variety of hyperbolic 3-manifolds
| Autor: | Stefano Francaviglia, Alessio Savini |
|---|---|
| Přispěvatelé: | Stefano Francaviglia, Alessio Savini |
| Rok vydání: | 2020 |
| Předmět: |
Mathematics - Differential Geometry
Ideal point Fundamental group 010102 general mathematics Holonomy Geometric Topology (math.GT) 57M50 53C24 22E40 01 natural sciences Character variety Hyperbolic volume Theoretical Computer Science Combinatorics Mathematics - Geometric Topology Mathematics (miscellaneous) Rigidity (electromagnetism) Differential Geometry (math.DG) Hyperbolic set Bounded function 0103 physical sciences FOS: Mathematics Volume of representations rigidity character varity 010307 mathematical physics Representations hyperbolic volume rigidity 0101 mathematics Mathematics |
| Zdroj: | ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE. :1325-1344 |
| ISSN: | 2036-2145 0391-173X |
| DOI: | 10.2422/2036-2145.201709_010 |
| Popis: | Given the fundamental group $\Gamma$ of a finite-volume complete hyperbolic $3$-manifold $M$, it is possible to associate to any representation $\rho:\Gamma \rightarrow \text{Isom}(\mathbb{H}^3)$ a numerical invariant called volume. This invariant is bounded by the hyperbolic volume of $M$ and satisfies a rigidity condition: if the volume of $\rho$ is maximal, then $\rho$ 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 $\Gamma$ into $\text{Isom}(\mathbb{H}^3)$ satisfies $\lim_{n \to \infty} \text{Vol}(\rho_n) = \text{Vol}(M)$, then there must exist a sequence of elements $g_n \in \text{Isom}(\mathbb{H}^3)$ such that the representations $g_n \circ \rho_n \circ g_n^{-1}$ converge to the holonomy of $M$. In particular if the sequence $\rho_n$ converges to an ideal point of the character variety, then the sequence of volumes must stay away from the maximum. We conclude by generalizing the result to the case of $k$-manifolds and representations in $\text{Isom}(\mathbb H^m)$, where $m\geq k$. Comment: 21 pages |
| Databáze: | OpenAIRE |
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