Combinatorial mapping-torus, branched surfaces and free group automorphisms

Gautero, François

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 6 (2007) no. 3, pp. 405-440.

We give a characterization of the geometric automorphisms in a certain class of (not necessarily irreducible) free group automorphisms. When the automorphism is geometric, then it is induced by a pseudo-Anosov homeomorphism without interior singularities. An outer free group automorphism is given by a $1$-cocycle of a $2$-complex (a standard dynamical branched surface, see [7] and [9]) the fundamental group of which is the mapping-torus group of the automorphism. A combinatorial construction elucidates the link between this new representation (first introduced in [16]) and the classical representation of a free group automorphism by a graph-map [2].

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