A train track directed random walk on Out(F-r)

作者：Kapovich Ilya^*; Pfaff Catherine

来源：International Journal of Algebra and Computation, 2015, 25(5): 745-798.

DOI：10.1142/S0218196715500186

• Fully irreducible
• free group automorphisms
• train track maps
• random walks
• index theory

Several known results, by Rivin, Calegari-Maher and Sisto, show that an element phi(n) is an element of Out(F-r), obtained after n steps of a simple random walk on Out(F-r), is fully irreducible with probability tending to 1 as n -> infinity. In this paper, we construct a natural "train track directed" random walk W on Out(Fr) (where r >= 3). We show that, for the element phi(n) is an element of Out(F-r), obtained after n steps of this random walk, with asymptotically positive probability the element phi(n) has the following properties: phi(n) is an ageometric fully irreducible, which admits a train track representative with no periodic Nielsen paths and exactly one nondegenerate illegal turn, that phi(n) has "rotationless index" 3/2 - r (so that the geometric index of the attracting tree T-phi n of phi(n) is 2r - 3), has index list {3/2 - r} and the ideal Whitehead graph being the complete graph on 2r - 1 vertices, and that the axis bundle of phi(n) in the Outer space CVr consists of a single axis.

• 出版日期2015-8