相对双曲群的边界理论及应用

In last 20 years, the theory of relatively hyperbolic groups has found important applications in the fields of low-dimensional topology and geometry, notable in the Agol and Wise’s solution of Virtual Haken / Fibering conjectures, and Dahmani, Guiradell and Osin’s work on the existence of purely pseudo-Anosov free normal subgroups in mapping class groups. Through a boundary point of view, this proposal aims to investigate the following problems on various boundaries of relatively hyperbolic groups:.1..Identification of the Hausdorff dimension of Floyd boundary with growth rate by a study of Patterson-Sullivan measures;.2..Understand Martin boundary by constructing a continuous surjective map from Martin boundary to Floyd boundary, and by characterizing the kernel of the map: this shall be achieved by establishing a relative version of Ancona inequality;.3..Measure the size of the contracting boundary introduced by Charney and Sultan in the visual boundary of CAT(0) groups with rank-1 elements: we shall prove it is null set by making use of growth tightness..These research problems will advance the understanding of relatively hyperbolic groups, CAT(0) groups with rank-1 elements and mapping class groups as well.

过去20年来，相对双曲群的理论在低维拓扑和几何中有许多重要的应用，特别体现在Agol和Wise解决几乎Haken/Fiber猜想，及Dahmani, Guiradell 和Osin在映射群中首次构造自由的纯粹pseudo-Anosov构成的正规子群等工作中。利用群的边界这一研究视角，本项目计划探讨如下三个与边界相关的研究问题：1. 通过研究边界上的Patterson-Sullivan测度，建立Floyd边界的Hausdorff维数和群增长率之间的等式；2.建立一个从Martin边界到Floyd边界的连续满射，通过刻画该商映射来理解Martin边界；3.比较Charney和Sultan引入的收缩边界与视觉边界的大小，来研究带有rank-1元素的CAT(0)群。我们相信这些问题的进展不仅会深化相对双曲群的理解，而且也会促进对带有rank-1元素的CAT(0)群和映射类群的研究。

项目研究了相对双曲群边界的度量性质以及与随机游走Martin边界之间的关系。我们在这些研究中产生出来的方法以被用于映射类群及CAT（0）群的计数研究，以及在三维流形群逆凸子群的研究。我们的研究成果如下：.1..Floyd边界的Hausdorff维数等于群的指数增长率，肯定回答了Bourdon的猜想。.2..随机游走的Martin边界满射Floyd边界，且在Floyd边界的锥点集合上是单射。.3..证明了映射类群的伪Anosov元素在关于Teichmuller度量下是指数典型的。.4..完全刻画了三维流形基本群的强逆凸子群，并给出了与高度有限性的等价性。.我们的研究成果发表Invent. Math., Geometry & Topology, Math Annalen（2篇）等国际重要杂志上，其中Invent. Math.为基础数学领域广受认可的的国际顶尖杂志。