Automorphisms of free groups and outer space.

*(English)*Zbl 1017.20035This is a survey on recent results on the automorphism groups \(\operatorname{Aut}(F)\) and outer automorphism groups \(\text{Out}(F)\) of finitely generated free groups, concentrating mainly on results obtained by geometric methods, and in particular by studying actions of \(\operatorname{Aut}(F)\) and \(\text{Out}(F)\) on suitable geometric objects. Many of these recent methods and results are motivated by analogies with geometric techniques for arithmetic and mapping class groups: for example, in analogy with the actions of mapping class groups on Teichmüller spaces, there is a proper action of \(\text{Out}(F)\) on “outer space” which consists of equivalence classes of marked metric graphs. Also, train track techniques inspired by Thurston’s theory of train tracks for surfaces are used to model a single automorphism. The first object discussed in the paper is outer space and its properties (dimension, contractibility, closure and boundary, connections with actions on \(\mathbb{R}\)-trees). The second part of the survey discusses various algebraic properties of the automorphism groups such as presentations, virtual finiteness properties, homology and Euler characteristics, fixed subgroup of an automorphism, various types of subgroups (finite, Abelian, solvable subgroups and Tits alternative), relations to arithmetic and linear groups as well as to automatic and hyperbolic groups. The paper closes with lists of open problems and of 132 references.

Reviewer: Bruno Zimmermann (Trieste)

##### MSC:

20F65 | Geometric group theory |

20E05 | Free nonabelian groups |

20F28 | Automorphism groups of groups |

57M07 | Topological methods in group theory |