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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
Abstract

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.

Keywords
free group, random walk, outer space, free group automorphisms, train track maps
Mathematical Subject Classification
Primary: 20F65
Secondary: 37D99, 57M99
References
Publication
Received: 5 May 2019
Revised: 31 January 2021
Accepted: 28 February 2021
Published: 5 April 2022
Proposed: Walter Neumann
Seconded: Mladen Bestvina, Dmitri Burago
Authors
Ilya Kapovich
Department of Mathematics and Statistics
Hunter College of CUNY
New York, NY
United States
http://math.hunter.cuny.edu/ilyakapo/
Joseph Maher
Department of Mathematics
CUNY College of Staten Island and CUNY Graduate Center
Staten Island, NY
United States
http://www.math.csi.cuny.edu/~maher/
Catherine Pfaff
Department of Mathematics and Statistics
Queen’s University
Kingston, ON
Canada
https://mast.queensu.ca/~cpfaff/
Samuel J Taylor
Department of Mathematics
Temple University
Philadelphia, PA
United States
https://math.temple.edu/~samuel.taylor/