After studying the first section, especially the section 1.3 (covering spaces) in Hatcher's book, it is obvious that many classical topics in group theory, especially in infinite group theory, have nice geometric interpretation, I can not help asking the following question:
Question 1: Are there any results in group theory that are much easier to be obtained by geometric, or topolocial tools that by purely algebraic tools?
Of course, the first typical example comes to mind is that any subgroup of a free group is free. Are there any other examples?
Since it is a common idea in group alegbras that we compare properties of groups with the algebra it generated, and we know that free group $F_2=Z*Z$ has a nice geometric interpretation by considering its cayley graph, I would like to ask the following question,
Question 2: Denote $L(F_2)$ to be the von-Neumann algebra associated to the free group generated by two elements, is there any geometric interpertation for this algebra?
Of course, another motivation to ask the 2nd question is that we have already many known candidate invariants associated to these free group factors, such as the free dimension in free probability, the number of generators etc, but in my opinion, most of them are more of less algebraic or analytic in natural.