Volume 26, issue 1 (2022)

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Random trees in the boundary of outer space

Ilya Kapovich, Joseph Maher, Catherine Pfaff and Samuel J Taylor

Geometry & Topology 26 (2022) 127–162
DOI: 10.2140/gt.2022.26.127

We prove that for the harmonic measure associated to a random walk on Out (Fr) satisfying some mild conditions, a typical tree in the boundary of Outer space is trivalent and nongeometric. This result answers a question of Mladen Bestvina.

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free group, random walk, outer space, free group automorphisms, train track maps
Mathematical Subject Classification
Primary: 20F65
Secondary: 37D99, 57M99
Received: 5 May 2019
Revised: 31 January 2021
Accepted: 28 February 2021
Published: 5 April 2022
Proposed: Walter Neumann
Seconded: Mladen Bestvina, Dmitri Burago
Ilya Kapovich
Department of Mathematics and Statistics
Hunter College of CUNY
New York, NY
United States
Joseph Maher
Department of Mathematics
CUNY College of Staten Island and CUNY Graduate Center
Staten Island, NY
United States
Catherine Pfaff
Department of Mathematics and Statistics
Queen’s University
Kingston, ON
Samuel J Taylor
Department of Mathematics
Temple University
Philadelphia, PA
United States