Length spectrum compactification of the $\mathrm{SO}_{0}(2,3)$-Hitchin component

Ouyang, Charles; Tamburelli, Andrea

doi:10.48550/arxiv.2010.03499

Cited by 3 publications

(8 citation statements)

References 29 publications

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“…In general the map − − → Mix(Σ) → Mix(Σ) has fibers (S 1 ) k where k denotes the number of connected components of Σ ′ . In [DLR10] the set Mix(Σ) is identified with the corresponding set of geodesic currents, a perspective that is generalized in [OT21b,OT20], where the flat metrics associated to cubic (resp. quartic) differentials are considered.…”

Section: We Denote Bymentioning

confidence: 99%

“…A more analytic, and independent approach to compactifications of the space of maximal representations in rank 2 Hermitian Lie groups has been independently pursued by Ouyang [Ouy19], Ouyang-Tamburelli [OT20,OT21a], and Martone-Ouyang-Tamburelli [MOT21]. In their work, they consider the length spectrum of the negatively curved metric induced on the unique invariant minimal surface associated to a maximal representation.…”

Section: Introductionmentioning

confidence: 99%

“…See Remark 5.6 in §5.2 for more details as well as a comparison with the structures considered in[DLR10] and in[OT21b,OT20] …”

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confidence: 99%

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Weyl chamber length compactification of the ${\rm PSL}(2,\mathbb R)\times{\rm PSL}(2,\mathbb R)$ maximal character variety

Burger¹,

Iozzi²,

Parreau³

et al. 2021

Preprint

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We study the vectorial length compactification of the space of conjugacy classes of maximal representations of the fundamental group Γ of a closed hyperbolic surface Σ in PSL(2, R) n . We identify the boundary with the sphere P((ML) n ), where ML is the space of measured geodesic laminations on Σ. In the case n = 2, we give a geometric interpretation of the boundary as the space of homothety classes of R 2 -mixed structures on Σ. We associate to such a structure a dual tree-graded space endowed with an R 2 + -valued metric, which we show to be universal with respect to actions on products of two R-trees with the given length spectrum.PSL(2, R) × PSL(2, R) MAXIMAL REPRESENTATIONS 5.1. Flat structures 18 5.2. R 2 -mixed structures on a surface 20 5.3. The R 2 -length function of a mixed structure 22 5.4. Interpretation of ML × ML as − − → Mix(Σ) 22 6. The R 2 -tree-graded space dual to a mixed structure 25 6.1. Generalities on R n -tree-graded spaces. 25 6.2. Construction of the R 2 -space X M 26 6.3. Basic properties of X M 28 7. Embeddings in products of trees 31 References 33

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Section: We Denote Bymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Weyl chamber length compactification of the ${\rm PSL}(2,\mathbb R)\times{\rm PSL}(2,\mathbb R)$ maximal character variety

Burger¹,

Iozzi²,

Parreau³

et al. 2021

Preprint

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show abstract

“…In particular, Gothen ( [Got01]) showed that there are precisely 3 • 2 2g + 2g − 4 connected components of maximal representations, of which only 2 2g + 2g − 3 are smooth. Of these, 2 2g are isomorphic copies of the Sp(4, R)-Hitchin component, which were analyzed in a previous work ([OT20a]). The remaining 2g − 3 are referred to as Gothen components in the literature and are the main object of study of this paper.…”

Section: Introductionmentioning

confidence: 99%

“…In a previous paper ( [OT20a]), we used this pseudo-Riemannian point of view to define the length spectrum of an Sp(4, R)-Hitchin representation and describe the boundary of this component as projectivized mixed structures, that is hybrid geometric objects on the surface that are measured laminations on some collection of incompressible subsurfaces and flat metrics of finite area induced by meromorphic quartic differentials on the complement. In this note, we show that all Gothen components share the same boundary, precisely Theorem A.…”

Section: Introductionmentioning

confidence: 99%