Random outer automorphisms of free groups: Attracting trees and their singularity structures

Kapovich, Ilya; Maher, Joseph; Pfaff, Catherine; Taylor, Samuel

doi:10.1090/tran/8472

Random outer automorphisms of free groups: Attracting trees and their singularity structures
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by Ilya Kapovich, Joseph Maher, Catherine Pfaff and Samuel J. Taylor PDF

Trans. Amer. Math. Soc. 375 (2022), 525-557 Request permission

Abstract:

We prove that a “random” free group outer automorphism is an ageometric fully irreducible outer automorphism whose ideal Whitehead graph is a union of triangles. This means that its attracting (and repelling) tree is a nongeometric $\mathbb {R}$-tree all of whose branch points are trivalent. In particular, it is not the dual $\mathbb {R}$-tree to a foliation on a finite simplicial 2-complex. This answers a question of Handel and Mosher.

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