Hyperbolic volume estimates via train tracks

TL;DR

This thesis develops a method to estimate hyperbolic volumes of 3-manifolds using train track splitting sequences, connecting surface dynamics with geometric invariants through quasi-isometric graph models.

Contribution

It introduces a novel approach to estimate distances in the pants graph via train track sequences and relates these estimates to hyperbolic volume bounds for mapping tori and related 3-manifolds.

Findings

Distance estimates are valid up to constants for train track splitting sequences.

The method provides volume bounds for mapping tori of pseudo-Anosov maps.

Applications include volume estimates for complements of closed braids.

Abstract

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.