1
$\begingroup$

Let $F$ be a hyperbolic surface of finite type. Suppose $\alpha$ is a simple closed geodesic and $\beta$ is any closed geodesic intersecting $\alpha$. Consider a Fenchel-Nielsen coordinate of the Teichmuller space containing $\alpha.$ Let $l_{x}$ denote the length of a geodesic $x$.

By "Margulis Lemma", there exists $\epsilon$ such that after $l_{\alpha}<\epsilon$, if we decrease $l_{\alpha}$, $l_{\beta}$ will increase exponentially.

Q) What happens to $l_{\beta}, $if we start increasing the length of $l_{\alpha}$?

More precisely, when we just increase $l_{\alpha}$, given any $L_1$ does there exists $L_2$ such that $l_{\beta}<L_2$, whenever $l_{\alpha}>L_1$. If so, can we explicitely find a relation between $L_1$ and $L_2$, i.e. what happens to $l_{\beta}$ in a continuous path in the Teichmüller space along which $l_{\alpha}$ increases.

Improve this question
$\endgroup$
9
  • $\begingroup$ This seems unlikely to hold as stated. For example if you take a third simple curve which intersects $\beta$ but not $\alpha$ and do Dehn twists around it then you get a sequence of pairwise isometric hyperbolic metrics in Teichmüller space where $l_\alpha$ is (any) constant and $l_\beta$ goes to infinity. $\endgroup$
    – Jean Raimbault
    Commented May 9, 2016 at 6:14
  • $\begingroup$ My question was, what happens if we increase the length of "$\alpha$", not the other way around. $\endgroup$
    – Cusp
    Commented May 9, 2016 at 6:43
  • $\begingroup$ yes, and if what I said above holds then for any value of $l_\alpha$ there are points in Teichmüller space where $l_\beta$ is as large as you want it to be, in particular there cannot be an upper bound on $l_\beta$ depending only on $l_\alpha$. $\endgroup$
    – Jean Raimbault
    Commented May 9, 2016 at 8:12
  • $\begingroup$ @JeanRaimbault Now I understand your point. I have changed the question a little bit. I want the relation when $l_{\alpha}$ increases. Thanks for pointing that out. $\endgroup$
    – Cusp
    Commented May 9, 2016 at 8:48
  • 1
    $\begingroup$ Another possibility, I guess closer to what you want, would be to look at what happens in a continuous path in Teichmüller space along which $l_\alpha$ increases. $\endgroup$
    – Jean Raimbault
    Commented May 9, 2016 at 9:47

0

Reset to default

You must log in to answer this question.