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The compression body graph has infinite diameter

Joseph Maher and Saul Schleimer

Algebraic & Geometric Topology 21 (2021) 1817–1856
Abstract

We show that the compression body graph, which is Gromov hyperbolic, has infinite diameter. Furthermore, every subgroup in the Johnson filtration of the mapping class group contains elements which act loxodromically on the compression body graph. Our methods give an alternative proof of a result of Biringer, Johnson and Minsky: the stable lamination of a pseudo-Anosov element is contained in the limit set of a compression body if and only if some power of the pseudo-Anosov element extends over a nontrivial subcompression body. We also extend results of Lubotzky, Maher and Wu, on the distribution of Casson invariants of random Heegaard splittings, to a larger class of random walks.

Keywords
curve complex, disc set, handlebody, compression body
Mathematical Subject Classification 2010
Primary: 37E30
Secondary: 20F65, 57M50
References
Publication
Received: 7 October 2019
Revised: 3 July 2020
Accepted: 19 July 2020
Published: 18 August 2021
Authors
Joseph Maher
Department of Mathematics
CUNY College of Staten Island and CUNY Graduate Center
Staten Island, NY
United States
Saul Schleimer
Mathematics Institute
University of Warwick
Coventry
United Kingdom