I'm trying to learn some stuff about divergence of geodesics. Let $\gamma$ be a geodesic in a metric space $X$. The divergence of $\gamma$ is a function $f(r)$ for $r \ge 0$ such that $f(r)$ is the length of a shortest path between $\gamma(-r)$ and $\gamma(r)$ in the complement $X \setminus B(\gamma(0), r)$ of a ball at $\gamma(0)$.

The first question I wanted to ask is the following:

Is there some geodesic in mapping class groups such that the divergence of it is at least quadratic?

It is known (due to Behrstock) that the orbit of a pseudo-anasov element is a quasi-geodesic $p$ and satisfying Morse property. Morse property of a quasi-geodesic $p$ means that any quasi-geodesic with endpoints at $p$ lies in a uniform neighborhood of $p$.

The second question is following:

Let $g$ be pA element. We connect $g^{-n}$ and $g^n$ by a geodesic $p_n$. Let $n \to \infty$. The $p_n$ will converge to a geodesic $p_\infty$ lying the uniform neighborhood of $\{g^n: n \in \mathbb N\}$. Does the geodesic $p_\infty$ have the quadratic divergence?

There seems lots of literatures about the divergence. If this is previously known, please be kind to tell me the reference.


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    $\begingroup$ The reference for your first question is this: arxiv.org/abs/math/0611359 . I am not sure what your second question is asking. $\endgroup$ – staylor Jul 18 '13 at 14:58
  • $\begingroup$ @staylor: Thanks a lot! I will have a look at it. I'm trying to re-formulate my question. But maybe your reference already answers it. $\endgroup$ – stephen Jul 18 '13 at 15:28

As Sam Taylor told you, the paper to read is by Duchin and Rafi. They prove that the divergence rate is between linear and quadratic. The second question is about rate of divergence along axes of pA elements in the mapping class group. In this case, the rate is quadratic, see theorem 4.2 in the same paper of Duchin and Rafi. This also answers the first question.

  • $\begingroup$ Misha, Thank you very much. I appreciate your time to point this out and give precise references. $\endgroup$ – stephen Jul 18 '13 at 19:37

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