Moduli of graphs and automorphisms of free groups

Finiteness of Mapping Class Groups

“…The Z-factor is characteristic, and is fixed by an index-2 subgroup of Out(F × Z). There is a surjection from this subgroup to Out(F ), and Out(F ) is virtually torsion-free by work of M. Culler and K. Vogtmann [6]. Since the kernel Hom(F, Z) ∼ = H 1 (F ) of this surjection is torsion-free, Out(F × Z) is also virtually torsion-free.…”

Stable homology of automorphism groups of free groups

“…Culler-Vogtmann's outer space plays the role for Out(F n ) that Teichmüller space plays for mapping class groups. Since its introduction in [CV86], it has been of central importance in the field, and firmly connects Out(F n ) to the study of graphs. A point in outer space X n is given by a triple (G, g, h), where G is a connected finite graph, g is a metric on G, i.e.…”

“…In particular A 0 n = Out(F n ) and A 1 n = Aut(F n ). BG 0 and BG 1 are directly related to automorphisms of free groups, via Culler-Vogtmann's outer space [CV86]. As mentioned in Section 1.2, outer space is a contractible space with an action of Out(F n ).…”

Cohomological structure of the mapping class group and beyond

“…In 1986, Culler and Vogtmann [11] defined a space X n , called the Outer Space, which plays the role of the Teichmüller space where the mapping class group is replaced by Out F n . In particular, they proved that X n is contractible and Out F n acts on it properly discontinuously.…”