作为几何群论领域的一个分支，自由群及其自同构群的研究在国际上极为活跃，国内正起步追赶，本项目将进行这方面的工作．..项目的研究对象为自由群的几何自同构．它处于曲面自同胚与自由群自同构研究的交叉领域，是一个富有内涵的课题．..本项目将分析..(1) 几何自同构的不动子群及它作用在自由群的Gromov边界的行为；..(2) 几何自同构在紧化Outer Space以及free factor复形上的作用．..最后用R-tree和lamination的语言给出几何自同构的内蕴描述，并构造算法判断自同构是否是几何的．..这将深化人们对带边紧曲面的映射类群在自由群的外自同构群中存在方式的认知．

As a branch of the field of geometric group theory, the research of free groups with their automorphism groups is active in the world. This project is proposed for catching the new trend of this field...We will focus on the topic of geometric automorphisms of free groups which connects the research of automorphisms of free groups with that of self homeomorphisms of surfaces. It is indeed a fruitful topic...Through analysing..(1) the fixed subgroups of geometric automorphisms and their actions on the Gromov boundary of free groups,..(2) the actions of geometric automorphisms on the compacted Outer Space and the free factor complex,..we will use the language of both "R-tree" and "lamination" to describe the sufficient and necessary conditions of an automorphism being geometric.