On differentiable compactifications of the hyperbolic plane and algebraic actions of SL(2;R) on surfaces

On differentiable compactifications of the hyperbolic space

The group of direct isometries of the real n-dimensional hyperbolic space is G=SOo(n,1). This isometric action admits many differentiable compactifications into an action on the closed ball. We prove that all such compactifications are topologically conjugate but not necessarily differentiably conjugate. We give the classifications of real analytic and smooth compactifications.

Kobayashi双曲复流形的度量紧化与Denjoy--Wolff定理

The metric compactification of a Kobayashi hyperbolic complex manifold and a Denjoy--Wolff Theorem

该文研究Kobayashi双曲复流形的度量紧化性质，证明其紧化的真实性与良好性，给出拟等距嵌入连续延拓判据，关联边界点纤维并表述Wolff型引理，证明满足边界发散条件的完备双曲区域的Denjoy--Wolff定理，还给出两类紧化间恒等映射连续延拓的充要条件。

连续可微度量与权重的洛伦兹分裂定理

A Lorentzian splitting theorem for continuously differentiable metrics and weights

本文通过椭圆技术与线适应曲线，将洛伦兹分裂定理推广至$C^1$度量与权重的全局双曲加权时空。

Tropical Geometric Compactification of Moduli, I - $M_g$ case -

We compactify the classical moduli variety of compact Riemann surfaces by attaching moduli of (metrized) graphs as boundary. The compactifications do not admit the structure of varieties and patch together to form a big connected moduli space in which $\sqcup_{g} M_{g}$ is open dense. The metrized graphs, which are often studied as "tropical curves", are obtained as Gromov-Hausdorff collapse by fixing diameters of the hyperbolic metrics of the Riemann surfaces. This phenomenon can be also seen as an archemidean analogue of the tropicalization of Berkovich analytification of $M_{g}$ (cf., [ACP]…

Linear algebra and unification of geometries in all scales

We present an idea of unifying small scale (topology, proximity spaces, uniform spaces) and large scale (coarse spaces, large scale spaces). It relies on an analog of multilinear forms from Linear Algebra. Each form has a large scale compactification and those include all well-known compactifications: Higson corona, Gromov boundary of hyperbolic spaces, the visual boundary of CAT(0)-spaces, \v Cech-Stone compactification, Samuel-Smirnov compactification, and Freudenthal compactification. As an application we get simple proofs of results generalizing well-known theorems from coarse topology. A …

Topological boundaries of connected graphs and Coxeter groups

We introduce and study certain topological spaces associated with connected rooted graphs. These spaces reflect combinatorial and order theoretic properties of the underlying graph and relate in the case of hyperbolic graphs to Gromov's hyperbolic compactification. They are particularly tractable in the case of Cayley graphs of finite rank Coxeter groups. In that context we speak of the compactification and the boundary of the Coxeter group. As it turns out, the canonical action of the Coxeter group on its Cayley graph induces a natural action on the compactification and the boundary. From thi…