Twisting Out Fully Irreducible Automorphisms

Abstract

By a theorem of Thurston, in the subgroup of the mapping class group generated by Dehn twists around two curves which fill, every element not conjugate to a power of one of the twists is pseudo-Anosov. We prove an analogue of this theorem for the outer automorphism group of a free group.

Current twisting and nonsingular matrices

“…With these notions we have shown the following analog to a classical theorem of Thurston: Key to our analysis in [12] and Section 5 of the present paper is the following theorem, which measures how the free volume changes upon twisting.…”

“…These bounds are shown in [12] to hold not only for cyclic subgroups, but for any finitely generated malnormal subgroup of F k ; in particular any proper free factor of F k . We will also need the following notions from [12] for Section 5.…”

“…Theorem 2.5 ( [12], Theorem 4.6). Let δ 2 be a Dehn twist automorphism corresponding to a very small cyclic tree T 2 with cyclic edge generator c 2 , and let T 1 be any other very small cyclic tree.…”

“…Let T 1 be a basis for F k relative to T 1 . Denote by φ n = δ [12] shows that since n is sufficiently large, for m ≥ 0 we have…”

“…To build such elements we apply a construction from our earlier work [12]; namely, we use Dehn twist automorphisms to build customized hyperbolic fully irreducible elements of Out F k . Satisfying the hypothesis then requires that we understand the stable and unstable currents in PCurr(F k ) associated to a product of Dehn twists.…”