六边形分解图的粗几何

The space of measured foliations of the hexagon

The theory of geometric structures on a surface with nonempty boundary can be developed by using a decomposition of such a surface into hexagons, in the same way as the theory of geometric structures on a surface without boundary is developed using the decomposition of such a surface into pairs of pants. The basic elements of the theory for surfaces with boundary include the study of measured foliations and of hyperbolic structures on hexagons. It turns out that there is an interesting space of measured foliations on a hexagon, which is equipped with a piecewise-linear structure (in fact, a na…

Coarse property {C} and decomposition complexity

The coarse category was established by Roe to distill the salient features of the large-scale approach to metric spaces and groups that was started by Gromov. In this paper, we use the language of coarse spaces to define coarse versions of asymptotic property C and decomposition complexity. We prove that coarse property C implies coarse property A; we also show that these coarse versions share many of the features of their metric analogs such as preservation by products or unions.

六边形分解与Weil-Petersson度量

Hexagon decompositions and the Weil-Petersson metric

本文构造了从六边形图到增广Teichmüller空间的拟等距映射，并证明其宽度至多$\sqrt{g+n}\log(g+n)$，同时给出边界层的显式投影。

Decomposition complexity with respect to coarse properties

We formalize the concept of a family of metric spaces satisfying a coarse property uniformly and we generalize finite decomposition complexity of Erik Guentner, Romain Tessera, and Guoliang Yu. Of particular interest are results determining sufficient conditions for a metric space to satisfy Property A of Guoliang Yu.

Coarse free products

We define a notion of free product for coarse spaces that generalizes the corresponding notion of a free product for groups. We show that free products preserve coarse properties such as coarse property C, finite coarse decomposition complexity, and coarse property A. We also give an upper bound estimate on the dimension of a coarse free product in terms of the dimension of its factors.

自由群的裤子图

The Pants Graph of a Free Group

本文研究自由群的裤子分解及其构造的裤子图，证明其连通且无界。