Random mapping-torus volume and torsion growth limits

Consider random mapping tori whose monodromies are random words, and let the genus of the fiber tend to infinity. Let the expected growth rates refer to growth as a function of the monodromy word length, for the hyperbolic volume and for the logarithm of the torsion subgroup of first homology. Volume and torsion growth-limit conjecture. The expected growth rates of both quantities approach finite limits as the genus of the fiber tends to infinity. This is an empirical conjecture based on the observed behavior of random mapping tori; the existence and finiteness of these genus limits remain open.