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Here are some questions about the earthquake deformation of hyperbolic surface that I can't answer or find references.

I briefly recall the settings. Let's fix a closed surface $S$ with genus $g\geq 2$. A point $h$ in the Teichmuller space $\mathscr{T}$ of $S$ may be thinked of either (a) as a marked hyperbolic structure on $S$
or (b) as a conjugate class of representations $\pi_1(S)\rightarrow PSL(2,\mathbb{R})$.

For any simple closed curve $\alpha$ on $S$, the Fenchel-Nielson twist along $\alpha$ gives rise to a flow $\phi_\alpha^t$ on $\mathscr{T}$. Wolpert proved that $\phi_\alpha^t$ is an Hamiltonian flow with respect to the Weil-Petersson symplectic structure. We can describe the deformation in the representation level. Namely, take a representation $h:\pi_1(S)\rightarrow PSL(2,\mathbb{R})$ which defines a point in $\mathscr{T}$, if $\alpha$ is seperating, then $\pi_1(S)$ is the amalgamated product of two groups, and $\phi_\alpha^t(h)$ is a representation in which we modify restriction of $h$ on one of the two groups by conjugation.

Thurston's eathquake deformation is a generalization of the above construction where we take a geodesic lamination instead of simple closed curve.

Question 1: Is there any example of an explicit family of representations $h_t:\pi_1(S)\rightarrow PSL(2,\mathbb{R})$ which gives an earthquake deformation supported on some non-simple lamination?

Question 2: Is the earthquake flow Hamiltonian, say, generated by the "generalized length function" of laminations?

Question 3: Can one describe the limit of $\phi_\alpha^t(h)$, as a projectived measured lamination, when $t\rightarrow\pm\infty$ (at least when $\alpha$ is simple closed)? For example, is it a measured lamination supported on $\alpha$?

Remark to question 3: I'm not sure of this, it seems that the review of a paper of Bonahon suggests $\phi_\alpha^t$ extends to a non-trivial action on the Thurston boundary of $\mathscr{T}$. If it is true, then the limit in question 3 does not always exist.


Addendum: Now I realized that I made some conceptual mistakes about Question 3. Once we think of the earthquake flow intuitively as "horocycle flow" like on the upper half plan, the picture would be clear.

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2 Answers 2

up vote 3 down vote accepted

On Q1 I would guess that you can find explicit deformations corresponding to an earthquake path on a non-simple lamination in the quite special case of the punctured torus. You might find some help for instance in

MR0697067 (85d:32047) Waterman, Peter; Wolpert, Scott Earthquakes and tessellations of Teichmüller space. Trans. Amer. Math. Soc. 278 (1983), no. 1, 157–167.

The answer to Q2 is yes, the earthquake flow is the Hamiltonian flow of the length function. I'm not sure where this was first proved but you could check in Kerckhoff's paper on the Nielsen realization problem.

On Q3 I'm convinced that the earthquake path $\phi^t_\alpha(h)$ limits to $\alpha$, I believe you could prove it by understanding the asymptotic behavior of the length of closed curves on this 1-parameter family of metrics, using the tools in the Kerckhoff paper mentioned above.

As for your remark, as Igor Rivin mentiond, I don't believe the Bonahon paper that you cite states that the earthquake flow extends to the boundary as nicely.

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You would do well to study the work of Bill Goldman (e.g. THE COMPLEX-SYMPLECTIC GEOMETRY OF SL(2,C)-CHARACTERS OVER SURFACES, and the references therein to his own work). Also, you might want to read the actual Bonahon paper, instead of the review. Also "I'd like some comments" is not an auspicious start to a question.

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This would be more appropriate as a comment rather than an answer. –  S. Carnahan Oct 4 '11 at 4:05
    
@Igor: Thanks, i've edited the beginning of the text. –  Xin Nie Oct 4 '11 at 5:17
    
@S. Carnahan: true, I thought that just after I wrote it... –  Igor Rivin Oct 4 '11 at 8:30

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