Geometry and Dynamics in Surfaces and Free Group Extensions

对象的对称性的代数属性由称为群的数学结构编码。从形式上讲，群只是一个简单的集合，它配备了一个二元运算，满足两个公理，形式化地定义了取消运算的概念。数学家既关心由这些规则产生的最重要的抽象代数结构，也关心具有自然意义的具体例子，如平面的刚性运动集、魔方的构形或多面体的对称性。在这里，自由群具有特别简单的代数结构，其中存在最小可能的消去，它扮演着构建许多其他群的构建块的重要角色。自由组扩展是这种构造的一个基本例子。然而，尽管是由众所周知的组件构建的，免费的群组扩展仍然相当神秘。这个项目将研究自由群扩张的动力学、代数和几何方面，并结合这些观点来阐明这些群的广泛结构。这扩展了研究人员与合作者先前的工作，探索和联系给定的群可以被建立为按循环自由的群的不同方式，以及关于某些自由群扩张的粗负曲率的工作。本项目将集中在自由群自同构和曲面同态的背景下的几何、拓扑、代数和动力学方面。这位研究人员将继续与合作者合作，通过2-络合物上某些半流的动力学来分析自由环基团的分裂。一个主要的目标是通过整个群在树上的均匀作用将不同分裂的所有单变量联系在一起，并利用这一点来定义正则代数不变量，并将所有的Cannon-瑟斯顿映射彼此关联。除了加强这些联系，该项目还将致力于证明具有完全不可约的单列的性质是由该群的所有分裂或没有分裂共享的。继续他与合作者的工作，这位研究人员将研究一般自由群扩张的几何，并刻画Gromov双曲性，并建立这类群的刚性结果。关于表面同胚的研究，研究人员将与合作者合作，证明所有小膨胀伪Anosov同态都具有对称性猜想所预测的简单结构。最后，研究者将研究曲面的TeichMuller空间的大规模几何和动力学性质，目的是计算三角形的期望厚度和曲面同胚自然集的格点渐近性。

The algebraic properties of the symmetries of an object are encoded by the mathematical structure known as a group. Formally, a group is simply a set equipped with a binary operation that satisfies two axioms formalizing the concept of undoing the operation. Mathematicians are concerned both with the overarching abstract algebraic structures that arise from these rules, and with concrete examples that have natural significance, such as the set of rigid motions of the plane, the configurations of a Rubik's cube, or the symmetries of a polyhedron. Here free groups, which have a particularly simple algebraic structure in which there is the minimal possible cancellation, play an essential role as building blocks from which many other groups are built. Free group extensions are a basic example of this sort of construction. Yet, despite being built from well-understood pieces, free group extensions remain rather mysterious. This project will investigate dynamical, algebraic, and geometric aspects of free group extensions and combine these perspectives to illuminate the broad structure of these groups. This expands on the investigator's prior work with collaborators exploring and relating the different ways a given group may be built as a free-by-cyclic group, as well as work on the coarse negative curvature of certain free group extensions.This project will focus on aspects of geometry, topology, algebra, and dynamics in the context of free group automorphisms and surface homeomorphisms. The investigator will continue his work with collaborators analyzing splittings of free-by-cyclic groups via the dynamics of certain semi-flows on 2-complexes. A main goal is to tie together all the monodromies of the different splittings via a uniformized action of the entire group on a tree and to use this to define canonical algebraic invariants and relate all the Cannon-Thurston maps to each other. In addition to strengthening these connections, the project will work to prove that the property of having a fully irreducible monodromy is shared by either all or none of the splittings of the group. Continuing his work with a collaborator, the investigator will study the geometry of general free group extensions and work to characterize Gromov hyperbolicity and establish rigidity results for this class of groups. With regard to the study of surface homeomorphisms, the investigator will work with collaborators to show that all small-dilatation pseudo-Anosov homeomorphisms have the simple structure predicted by the symmetry conjecture. Finally, the investigator will study large-scale geometric and dynamical properties of the Teichmuller space of a surface with an aim towards calculating expected thinness of triangles and lattice point asymptotics for natural sets of surface homeomorphisms.