The compression body graph has infinite diameter

Maher, Joseph; Schleimer, Saul

doi:10.2140/agt.2021.21.1817

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The compression body graph has infinite diameter

Joseph Maher and Saul Schleimer

Algebraic & Geometric Topology 21 (2021) 1817–1856

DOI: 10.2140/agt.2021.21.1817

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

Volume 21, issue 4 (2021)