通过训练轨道的双曲线体积估计
Hyperbolic volume estimates via train tracks
摘要：
在这篇论文中，我们描述了如何估计裤子图中的距离跨越的训练轨道分裂序列的表面上，乘法和加法常数。如果对分裂序列作适当的假设，其中的训练轨道的每个顶点集将表示一个图的顶点，该图与裤子图自然拟等距;此外，分裂序列给出了该图中的边路径，因此，更准确地说，我们的距离估计在该路径的极值点之间成立。本距离估计受到Masur的结果的启发，Mosher和Schleimer在标记图中的距离。然而，我们只能在对分裂序列进行一些操作之后才能应用他们的证明路线：重新安排，改变基本移动的执行顺序，以便将产生Dehn扭曲的移动放在一起;然后进行解扭，该解扭抑制这些后一种移动中的大多数以给出新的序列，该序列不以与之前相同的轨道结束，但不包括裤子图中几乎静止的任何部分。然后，所需的距离是在解扭序列中发生的分裂的数量，直到常数。我们的主要定理的一个结果与Brock的一个结果一起是，给定一个表面$S$的伪Anosov自同构$ψ$，由Agol引入的最大分裂序列给出了由$S$和$ψ$建立的映射环面的双曲体积的估计。也有一些有趣的后果双曲体积的固体环面减去一个封闭的辫子，通过一个机器所采用的Dynnikov和Wiest。
英文摘要：
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 $ψ$ 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 $ψ$. 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
