Linking numbers of measured foliations

拟Fuchsian流形无穷远处的可测叶状结构

Measured foliations at infinity of quasi-Fuchsian manifolds

本文研究拟Fuchsian流形无穷远边界上的可测叶状结构对，证明了接近Fuchsian情形时该叶状结构对具有填充性，并进一步证明任意填充对在充分小的尺度下均可实现为某拟Fuchsian流形的无穷远叶状结构，从而回答了Schlenker在Fuchsian轨迹附近提出的问题。

Meromorphic quadratic differentials and measured foliations on a Riemann surface

We describe the space of measured foliations induced on a compact Riemann surface by meromorphic quadratic differentials. We prove that any such foliation is realized by a unique such differential $q$ if we prescribe, in addition, the principal parts of $\sqrt q$ at the poles. This generalizes a theorem of Hubbard and Masur for holomorphic quadratic differentials. The proof analyzes infinite-energy harmonic maps from the Riemann surface to $\mathbb{R}$-trees of infinite co-diameter, with prescribed behavior at the poles.

Hyperbolic metrics, measured foliations and pants decompositions for non-orientable surfaces

We provide analogues for non-orientable surfaces with or without boundary or punctures of several basic theorems in the setting of the Thurston theory of surfaces which were developed so far only in the case of orientable surfaces. Namely, we provide natural analogues for non-orientable surfaces of the Fenchel-Nielsen theorem on the parametrization of the Teichmüller space of the surface, the Dehn-Thurston theorem on the parametrization of measured foliations in the surface, and the Hatcher-Thurston theorem, which gives a complete minimal set of moves between pair of pants decompositions of th…

Fuchsian轨迹附近拟Fuchsian流形的无穷远测度叶状结构与常平均曲率曲面

Measured foliations at infinity and constant mean curvature surface in quasi-Fuchsian manifolds close to the Fuchsian locus

本文研究接近Fuchsian轨迹的拟Fuchsian流形，证明小尺度填满型无理测度叶状结构可唯一实现为无穷远测度叶状结构，并证明近Fuchsian流形可由CMC曲面单调叶状化。

Infinite sequence of fixed point free pseudo-Anosov homeomorphisms

We construct infinite sequences of pseudo-Anosov homeomorphisms without fixed points and leaving invariant a sequence of orientable measured foliations on the same topological surface and the same stratum of the space of abelian differentials. The existence of such sequences show that all pseudo-Anosov homeomorphisms fixing orientable measured foliations cannot be obtained by the Rauzy-Veech induction strategy.

Notes on the Schwarzian tensor and measured foliations at infinity of quasifuchsian manifolds

The boundary at infinity of a quasifuchsian hyperbolic manifold is equiped with a holomorphic quadratic differential. Its horizontal measured foliation $f$ can be interpreted as the natural analog of the measured bending lamination on the boundary of the convex core. This analogy leads to a number of questions. We provide a variation formula for the renormalized volume in terms of the extremal length $\ext(f)$ of $f$, and an upper bound on $\ext(f)$. We then describe two extensions of the holomorphic quadratic differential at infinity, both valid in higher dimensions. One is in terms of Poinca…

映射类群的极限集

Limit sets of mapping class groups

该研究在射影测度叶状结构空间中为映射类群子群引入两类极限点概念，将Teichmüller测地线射线分类，且证明分类可由定向叶状结构的拓扑性质确定。

二面角球面曲面的模空间与测度叶状结构

Moduli Space of Dihedral Spherical Surfaces and Measured Foliations

本文研究二面角球面曲面，通过定义一对横向测度叶状结构来完全刻画该类曲面，并引入几何分解与形变方法，最终确定给定锥角和拓扑类型下模空间的维数。

带边曲面上测量叶状结构组合单位球的研究

Around the combinatorial unit ball of measured foliations on bordered surfaces

本文研究带边曲面上测量叶状结构组合单位球体积的Kontsevich可积性，确定了使该体积的s次幂可积的实数s范围，并揭示其结果依赖于曲面拓扑，与双曲曲面情形形成对比。

Classification des feuilletages moyennables par surfaces

We prove that the foliated Euler caracteristic classifies amenable measured foliations up to those defined by ergodic actions of the euclidian plane.

Convergence of some horocyclic deformations to the Gardiner-Masur boundary

We introduce a deformation of Riemann surfaces and we are interested in the convergence of this deformation to a point of the Gardiner-masur boundary of Teichmueller space. This deformation, which we call the horocyclic deformation, is directed by a projective measured foliation and belongs to a certain horocycle in a Teichmueller disc. Using works of Marden and Masur and works of Miyachi, we show that the horocyclic deformation converges if its direction is given by a simple closed curve or a uniquely ergodic measured foliation.

Closed/open string diagrammatics

We introduce a combinatorial model based on measured foliations in surfaces which captures the phenomenology of open/closed string interactions. The predicted equations are derived in this model, and new equations can be discovered as well. In particular, several new equations together with known transformations generate the combinatorial version of open/closed duality. On the topological and chain levels, the algebraic structure discovered is new, but it specializes to a modular bi-operad on the level of homology.

Arithmetic of hyperbolic 3-manifolds

This note is an elaboration of the ideas and intuitions of Grothendieck and Weil concerning the "arithmetic topology". Given 3-dimensional manifold M fibering over the circle we introduce an real quadratic number field K with discriminant d, where d>0 is an integer number uniquely determined by M. The idea is to relate geometry of M to the arithmetic of field K. On this way, we show that Vol M is a limit density of ideals of given norm in the field K (Dirichlet density). The second statement says that the number of cusp points of manifold M is equal to the class number of the field K. It is re…