  In this thesis we describe how to estimate the distance spanned in the pants
graph by a train track splitting sequence on a surface, up to multiplicative
and additive constants. If some moderate assumptions on a splitting sequence
are satisfied, each vertex set of a train track in it will represent a vertex
of a graph which is naturally quasi-isometric to the pants graph; moreover the
splitting sequence gives an edge-path in this graph so, more precisely, our
distance estimate holds between the extreme points of this path. The present
distance estimate is inspired by a result of Masur, Mosher and Schleimer for
distances in the marking graph. However, we can apply their line of proof only
after some manipulation of the splitting sequence: a rearrangement, changing
the order the elementary moves are performed in, so that the ones producing
Dehn twists are brought together; and then an untwisting, which suppresses the
majority of these latter moves to give a new sequence, which does not end with
the same track as before, but does not include any portion that is almost
stationary in the pants graph. The required distance is then, up to constants,
the number of splits occurring in the untwisted sequence. A consequence of our
main theorem together with a result of Brock is that, given a pseudo-Anosov
self-diffeomorphism $\psi$ of a surface $S$, the maximal splitting sequence
introduced by Agol gives us an estimate for the hyperbolic volume of the
mapping torus built from $S$ and $\psi$. There are also some interesting
consequences for the hyperbolic volume of a solid torus minus a closed braid,
via a machinery employed by Dynnikov and Wiest.
