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

自由群外自同构群的几何与动力学

• 批准号: 402300-2011
• 负责人: Pettet, Alexandra
• 金额: $ 1.17万
• 依托单位: University of British Columbia
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2012
• 资助国家: 加拿大
• 起止时间: 2012-01-01 至 2013-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=508646
• 关键词: Geometry; dynamics; outer; automorphism; group

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.

自由群的外自同构群Out（F）与曲面的映射类群Mod（S）的关系是研究自由群的外自同构群Out（F）的一个成功的方案。这两个群在数学的许多领域都扮演着重要的角色，包括代数几何、几何群论和低维拓扑;它们也参与了理论物理的一些最新发展。它们被认为是线性群的推广，并且确实与线性群共享许多性质。而线性组，即，由矩阵组成的群，已经得到了很好的研究，并出现在数学，物理和计算机科学中的任何地方。因为矩阵相对容易理解，所以当研究一个特定的群时，了解它与线性群的相似程度是很有成效的。我的研究是这个更大计划的一部分，该计划旨在了解Out（F）和Mod（S）成为线性群体的可能障碍，以及了解它们的行为在多大程度上类似于线性群体。我使用的方法是那些几何群论，工作的原则，即代数性质的群体反映在几何和动态的空间，他们的行动。