Geometric group theory and surface dynamics

该计划在二维动力系统和几何群论的相关领域分为三个项目。 每个项目都有不同的合著者，并建立在以前的联合工作基础上。 第一个是与约翰·弗兰克斯的合作。 一个目标是找到一个结构定理的熵零，面积保持同构的曲面。 另一个问题是判定亏格足够高的曲面的映射类群的有限指数子群是否能通过保面积的超同态忠实地作用在曲面上。 第二个项目的目标是与Mark Feighn合作，为自由群的外自同构群的共轭问题提供一个完整的解决方案。 第三个项目，这是一个与李莫舍合作，也解决了自由群的外部自同构群的基本性质;目标是证明子群分类定理的一个相对版本，并发展一个层次理论。该建议利用并进一步发展了曲面的映射类群之间的深层联系，曲面的外自同构群和自由群的外自同构群。关于正熵曲面的自同构，我们已经知道了很多。 该建议的一部分是发展零熵，面积保持表面仿射的结构定理。 本文利用相对映射类群技巧，研究了映射类群的有限指数子群在曲面上作用的存在性。 其他部分的建议，寻求推广已知的重要结果映射类组的外部自同构群的自由群。 在这些结果是共轭问题，一个相对版本的子群分类定理和分层理论：第一个要求的算法，决定如果两个元素不同，只有通过改变坐标;第二个是一个完整的模拟子群的一个基本分类定理的个别元素;第三部分探讨了与外自同构群相关的各种复形和空间的几何。

The proposal divides into three projects in the related fields of two dimensional dynamical systems and geometric group theory. Each project is with a different coauthor and builds on previous joint work. The first is a collaboration with John Franks. One goal is to find a structure theorem for entropy zero, area preserving diffeomorphism of surfaces. Another is to decide if a finite index subgroup of the mapping class group of a surface of sufficiently high genus can act faithfully on a surface by area preserving diffeomorphisms. The goal of the second project, which is a collaboration with Mark Feighn, is to provide a complete solution to the conjugacyproblem for the outer automorphism group of the free group. The third project, which is a collaboration with Lee Mosher, also addresses fundamental properties of the outer automorphism group of the free group; the goals are to prove a relative version of the subgroup classification theorem and to develop a heirarchy theory.The proposal makes use of, and further develops, the deep connections between the mapping class group of a surface, the diffeomorphism group of a surface and the outer automorphism group of the free group. A great deal is known about positive entropy surface diffeomorphisms. One part of the proposal is to develop a structure theorem for zero entropy, area preserving surface diffeomorphisms. This work makes use of relative mapping class group techniques and has applications to the existence of actions of finite index subgroups of mapping class groups on surfaces. Other parts of the proposal seek to generalize known important results about the mapping class group to the outer automorphism group of the free group. Among these results are the conjugacy problem, a relative version of the subgroup classification theorem and a heirarchy theory: the first asks for an algorithm that decides if two elements differ only by a change of coordinates; the second is a complete analogue for subgroups of a basic classification theorem for individual elements; the third explores the geometry of various complexes and spaces associated to the outer automorphism group.