Modular circle quotients and PL limit sets

Eisenstein-Picard模曲面的拓扑

The topology of the Eisenstein-Picard modular surface

该研究聚焦Eisenstein-Picard模曲面这一复双曲平面商空间，确定了其作为4-轨形的整体拓扑，推进了相关模曲面拓扑领域的拓扑结构研究。

格、Garside结构与弱模图

Lattices, Garside structures and weakly modular graphs

本文研究了具有 $\widetilde A_n$ 形单纯形的单纯复形的组合非正曲率性质，证明了具有保序 $\mathbb Z$ 作用的格及其格商均构成弱模图，并揭示了范畴 Garside 结构与格的关系，给出了具有奇异性质的 Garside 群首批实例。

通过基本群区分允许圆作用的2-结

Distinguishing 2-knots admitting circle actions by fundamental groups

本文研究3-球面中不同1-结与互质整数对构造的分支扭转旋转的唯一性问题，证明大多数情况下不存在产生相同分支扭转旋转的不同1-结，并利用循环型3-轨道流形的基本群作为补空间基本群的商群来区分分支扭转旋转。

The boundary of the deformation space of the fundamental group of some hyperbolic 3-manifolds fibering over the circle

By using Thurston's bending construction we obtain a sequence of faithful discrete representations ρ_n of the fundamental group of a closed hyperbolic 3-manifold fibering over the circle into the isometry group Iso H^4 of the hyperbolic space H^4. The algebraic limit of ρ_n contains a finitely generated subgroup F whose 3-dimensional quotient Ω(F)/F has infinitely generated fundamental group, where Ω(F) is the discontinuity domain of F acting on the sphere at infinity. Moreover F is isomorphic to the fundamental group of a closed surface and contains infinitely many conjugacy classes of maxima…

On conformally flat circle bundles over surfaces

We study surface groups $Γ$ in $SO(4,1)$, which is the group of Mobius tranformations of $S^3$, and also the group of isometries of $\mathbb{H}^4$. We consider such $Γ$ so that its limit set $Λ_Γ$ is a quasi-circle in $S^3$, and so that the quotient $(S^3 - Λ_Γ) / Γ$ is a circle bundle over a surface. This circle bundle is said to be conformally flat, and our main goal is to discover how twisted such bundle may be by establishing a bound on its Euler number. By combinatorial approaches, we have two soft bounds in this direction on certain types of nice structures. In this article we also const…

Surfaces, circles, and solenoids

Previous work of the author has developed coordinates on bundles over the classical Teichmueller spaces of punctured surfaces and on the space of cosets of the Moebius group in the group of orientation-preserving homeomorphisms of the circle, and this work is surveyed here. Joint work with Dragomir Saric is also sketched which extends these results to the setting of the Teichmueller space of the solenoid of a punctured surface, which is defined here in analogy to Dennis Sullivan's original definition for the case of closed surfaces. Because of relations with the classical modular group, the pu…

Weil--Petersson geodesics on the modular surface

We consider the Weil--Petersson (WP) metric on the modular surface. We lift WP geodesics to the universal cover of the modular surface and analyse geometric properties of the lifts as paths in the hyperbolic metric on the universal cover. For any pair of distinct points in the thick part of the universal cover, we prove that the WP and hyperbolic geodesic segments that connect the pair, fellow-travel in the thick part and all deviations between these segments arise during cusp excursions. Furthermore, we give a quantitative analysis of the deviation during an excursion. We leverage the fellow …

The Group of Quasisymmetric Homeomorphisms of the Circle and Quantization of the Universal Teichmüller Space

In the first part of the paper we describe the complex geometry of the universal Teichmüller space $\mathcal T$, which may be realized as an open subset in the complex Banach space of holomorphic quadratic differentials in the unit disc. The quotient $\mathcal S$ of the diffeomorphism group of the circle modulo Möbius transformations may be treated as a smooth part of $\mathcal T$. In the second part we consider the quantization of universal Teichmüller space $\mathcal T$. We explain first how to quantize the smooth part $\mathcal S$ by embedding it into a Hilbert-Schmidt Siegel disc. This qua…