TY - JOUR
T1 - A train track directed random walk on Out (Fr)
AU - Kapovich, Ilya
AU - Pfaff, Catherine
N1 - Publisher Copyright:
© 2015 World Scientific Publishing Company.
PY - 2015/8/18
Y1 - 2015/8/18
N2 - Several known results, by Rivin, Calegari-Maher and Sisto, show that an element φn Out(Fr), obtained after n steps of a simple random walk on Out(Fr), is fully irreducible with probability tending to 1 as n → ∞. In this paper, we construct a natural' train track directed' random walk on Out(Fr) (where r ≥ 3). We show that, for the element φn ∈ Out(Fr), obtained after n steps of this random walk, with asymptotically positive probability the element φn has the following properties: φn is an ageometric fully irreducible, which admits a train track representative with no periodic Nielsen paths and exactly one nondegenerate illegal turn, that φn has "rotationless index" 3/2-r (so that the geometric index of the attracting tree Tφ of φn 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 φn in the Outer space CVr consists of a single axis.
AB - Several known results, by Rivin, Calegari-Maher and Sisto, show that an element φn Out(Fr), obtained after n steps of a simple random walk on Out(Fr), is fully irreducible with probability tending to 1 as n → ∞. In this paper, we construct a natural' train track directed' random walk on Out(Fr) (where r ≥ 3). We show that, for the element φn ∈ Out(Fr), obtained after n steps of this random walk, with asymptotically positive probability the element φn has the following properties: φn is an ageometric fully irreducible, which admits a train track representative with no periodic Nielsen paths and exactly one nondegenerate illegal turn, that φn has "rotationless index" 3/2-r (so that the geometric index of the attracting tree Tφ of φn 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 φn in the Outer space CVr consists of a single axis.
KW - Fully irreducible
KW - free group automorphisms
KW - index theory
KW - random walks
KW - train track maps
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U2 - 10.1142/S0218196715500186
DO - 10.1142/S0218196715500186
M3 - Article
AN - SCOPUS:84946187775
SN - 0218-1967
VL - 25
SP - 745
EP - 798
JO - International Journal of Algebra and Computation
JF - International Journal of Algebra and Computation
IS - 5
ER -