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.
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