The compression body graph has infinite diameter

Joseph Maher, Saul Schleimer

Published 2018-03-16Version 1

We show that the compression body graph has infinite diameter, and that 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 alternate proof of a result of Biringer, Johnson and Minsky, that the stable and unstable laminations of a pseudo-Anosov element are contained in the limit set of a compression body if and only if some power of the pseudo-Anosov element extends over a non-trivial 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.