TY - JOUR

T1 - A train track directed random walk on Out (Fr)

AU - Kapovich, Ilya

AU - Pfaff, Catherine

N1 - Funding Information:
The first author was partially supported by the NSF Grant DMS-1405146 and by the Simons Foundation Collaboration Grant No. 279836. The second author was supported first by the ARCHIMEDE Labex (ANR-11-LABX-0033) and the A*MIDEX project (ANR-11-IDEX-0001-02) funded by the “Investissements d’Avenir” French government program managed by the ANR. She is secondly supported by the CRC701 grant of the DFG, supporting the projects B1 and C13 in Bielefeld. Both authors acknowledge support from U.S. National Science Foundation grants DMS 1107452, 1107263, 1107367 “GEAR Network”.
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

VL - 25

SP - 745

EP - 798

JO - International Journal of Algebra and Computation

JF - International Journal of Algebra and Computation

SN - 0218-1967

IS - 5

ER -