Geometry and dynamics of the outer automorphism group of a free group

基于自由群与曲面映射类群Mod(S)的关系，给出了一个研究自由群外部自同构群Out(F)的成功方案。这两个群体在数学的许多领域都发挥着重要的作用，包括代数几何、几何群论和低维拓扑；他们还讨论了理论物理学的一些最新发展。它们被认为是线性群的一般化，并且确实与线性群有许多共同的性质。反过来，线性群，即由矩阵组成的群，已经得到了很好的研究，并且在数学、物理和计算机科学中无处不在。由于矩阵相对容易理解，因此在研究一个特定的群以了解它与线性群的相似程度时，它是有效的。我的研究是理解Out(F)和Mod(S)作为线性群体可能存在的障碍，以及理解它们的行为在多大程度上与线性群体相似这一更大项目的一部分。我使用的方法是几何群论的方法，其原理是群的代数性质反映在它们所作用的空间的几何和动力学中。

A successful program for studying the outer automorphism group Out(F) of a free group has been based on its relationship to the mapping class group Mod(S) of a surface. Both groups play an important role in many areas of mathematics, including algebraic geometry, geometric group theory, and low-dimension topology; they have also entered into some recent developments in theoretical physics. They are thought to be generalizations of, and indeed share many properties with, linear groups. In turn linear groups, i.e., groups consisting of matrices, have been well studied and appear everywhere in mathematics, physics, and computer science. Because matrices are relatively easy to understand, it is productive when studying a particular group to understand how closely it resembles a linear group. My research is part of this larger program of understanding possible obstructions to Out(F) and Mod(S) being linear groups, as well as understanding the extent to which their behaviour nonetheless resembles that of linear groups. The methods I use are those of geometric group theory, working on the principal that the algebraic properties of groups are reflected in the geometry and dynamics of the spaces they act on.