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I have a question concerning limits of sequences of points in Teichmuller space, and how this notion is preserved under fellow-travelling.

Suppose that we have closed surface of genus $g\geq 2$, and a sequence $\{X_n\}_{n\in \mathbb{N}}$ of points in its Teichmuller space $T(S)$.

Suppose furthermore that, for $n\rightarrow \infty$ this sequence converges to a point $\lambda\in \mathbb{P}ML(S)$ in Thurston's compactification of Teichmuller space. For example, we can take a diverging sequence of points on a Teichmuller geodesic associated to a pseudo-Anosov automorphism of $S$.

Now suppose that we have another sequence $\{Y_n\}_{n\in \mathbb{N}}$ of points in $T(S)$ which fellow travels the first, meaning that $$d(X_n,Y_n)\leq r$$ for all $n$ and some $r>0$. Here, $d$ denotes the Teichmuller distance.

Here are the questions:

  1. Does the sequence $\{Y_n\}$ converge to some point in $\mathbb{P}ML(S)$?

  2. If (1) is true, is the limit point of $\{Y_n\}$ the same as the limit point of $\{X_n\}$?

It seems to me that the answer to both questions should be positive, but I am having trouble in proving the above statements, as convergence to a point in the boundary is expressed in terms of ratios between hyperbolic lengths and transverse measures of curves, while Teichmuller distance is related to the stretch factor of Teichmuller maps. These notions are (at least for me) not easily related.

Any help is kindly appreciated!

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2 Answers 2

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There are some limited situations where your question has a positive answer, which can be stated in the setting of "convex cocompact subgroups of the mapping class group" $\text{MCG}(S)$, as developed in a paper of myself and Farb with title as in the quotes.

Namely, if $\Gamma < \text{MCG}(S)$ is convex cocompact, and if $Z \in T(S)$ is any point, and if there exists $s>0$ such that the sequence $X_n$ stays within distance $s$ of the $\Gamma$-orbit of $Z$, then both of your questions 1 and 2 are answered affirmatively.

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You should consult Kasra Rafi's oevre, and many (if not all) things will be revealed.

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  • $\begingroup$ Thank you very much for the reference. It looks like the result which best approximates my question in Rafi's paper is theorem 7.1, together with remark 7.2. Maybe I am wrong, but it is not precisely an answer to my question. These theorem derive a criterion for fellow travelling, under the assumptions that the starting points lie in the thick part and are close enough. I assume fellow travelling between two sequences and ask whether their limits should coincide, provided they exist. Probably the techniques introduced by Rafi can be applied to answer my question. $\endgroup$
    – leone slavich
    Commented Mar 5, 2015 at 15:16

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