# Distance at least $n$ geodesics on surface

everyone. Let $S$ be a closed orientable 2-surface with genus at least 2. Let $\{C_{i}\mid i=1,...,3g-3\}$ be a pants decomposition of $S$. Suppose that $\{l_{C_{i}},\tau_{C_{i}}, for~ i=1,...,3g-3\}$ be a point in Fenchel Nielsen coordinates. Then $S$ admits a hyperbolic metric with respect to the point.

Now I want to know:

For any given integer $n\geq 2$, is there a way to classify any two geodesics with distance at least $n$ from the geometry, such as the length of these two geodesics?

Note: The distance defined on two closed geodesics is the distance in curve complex.
And I know that using pseudo-anosov map is a good way to construct distance at least $n$ two geodesics. But I really want to know is it possible to get this just by using the metric defined on the surface.

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I have no idea what you are asking. The first paragraph has nothing to do with the rest. The second paragraph is meaningless as stated. Do you want to get a lower bound (that's the best one can hope for) on the lengths of closed geodesics in terms of their distance in the curve complex? This one can do easily using the bound on the intersection number. The last paragraph again contains no meaningful questions. – Misha Jul 20 '13 at 4:00
I think you are asking something about how to compute distance (in the curve complex) between a) the systoles of a given geometry and b) a given simple closed geodesic curve. Furthermore you want this computation done just in terms of the geometry of the surface? If that is the case, let me point out that length, by itself, will only give an upper bound for distance. – Sam Nead Jul 20 '13 at 7:30
@SamNead: can you actually write down such an upper bound? – Igor Rivin Jul 20 '13 at 23:32
@Igor: I wrote such estimate in my 1992 paper on intersection pairing on hyperbolic manifolds as a toy case for a much harder 4-d problem. – Misha Jul 21 '13 at 2:03
@Igor - Suppose that $X$ is the given hyperbolic surface. Let $\alpha$ be the given geodesic curve. Let $\omega$ be the systole of $X$ - the shortest simple closed geodesic in $X$. We must give a bound on the distance between $\alpha$ and $\omega$ in the curve complex of $S$, the underlying topological surface. Since $\omega$ is the systole, we get an upper bound on its length and so a lower bound on the radius of its collar. We can now bound the intersection number between $\alpha$ and $\omega$ from above, solely in terms of the length of $\alpha$. Turning an upper bound on ... – Sam Nead Jul 21 '13 at 20:50