Sujet de Thèse
• Titre : Counting curves in hyperbolic surfaces
• Unité de recherche : IRMAR, UMR-6625
• Thème : Geometry, dynamics, low-dimensional topology
• Mots clefs : Hyperbolic geometry, Teichmüller space, mapping class group, ergodicity, rigidity
• Les noms, prénoms, courriel, établissement des directeurs ou directrices de thèse
1. Juan Souto, jsouto@gmail.com, IRMAR
Objectif de la thése
The goal of this project is to study variations of a result of Maryam Mirza- khani. She proved namely that if Σ is a orientable hyperbolic surface of genus gand with ncusps then the limit
L→∞lim 1
L6g−6+2n|{γ closed geodesic of typeγ0 with length`Σ(γ)≤L}|
exists and is positive for any homotopically essential curve γ0. Here two curves are of the same type if they differ by a mapping class or, in more mundane terms, if the associated free homotopy classes differ by a self- diffeomorphisms of Σ. Mirzakhani proved the existence of the limit above for simple curves in her thesis, and it is probably fair to say that this result played a role in her being awarded the Fields Medal in 2014. She dealt with the case of non-simple curves in 2016, her last paper, and lately there has been quite a bit of activity, both about the simple case as about the general one. A problem that is still widely open is what happens when the mapping class group is replaced by one of its subgroups.
Problem: Let G ⊂ Map(Σ) be a subgroup of the mapping class group.
Determine the asymptotic behaviour when L → ∞ of the number of curves γ ∈G·γ0 with `Σ(γ)≤L.
This problem is fully understood if G ⊂ Map(Σ) has finite index, and, although we didn’t finish writing the paper yet, the case thatGis the funda- mental group of a Veech curve in moduli space seems also settled. But that
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seems to be all, and there are plenty of subgroups Gfor which the problem above would be interesting: the centraliser of a subgroup ofMap(Σ), the im- ages of Prym representations, the Torelli group, the images inMap(S)of the fundamental groups of the strata the moduli space of abelian or quadratic differentials, Schottky subgroups ofMap(Σ)... Each one of these cases would be interesting and challenging, with some new insight needed in all cases.
The plan is then to start working on the problem above in the case that G is the centraliser of a subgroup of Map(Σ).
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