# Hyperbolic length of curve that does not enter a collar

Let $$\Sigma$$ be a compact surface of genus at least $$1$$ with one boundary component, equipped with a hyperbolic metric so that the boundary is geodesic. There is a version of the collar lemma that says there is a collar neighbourhood $$C$$ of the boundary such that no simple closed geodesic on $$\Sigma$$ enters $$C$$.

Given a curve, I am interested in measuring what length of it lies outside $$C$$. More precisely, I am curious about the following: does there exist a constant $$d>0$$ such that, if $$\gamma$$ is an essential simple closed curve (not nullhomotopic and not homotopic to the boundary component), then $$\ell(\gamma|_{\Sigma\backslash C})+d \geq \ell(\tilde{\gamma})$$ Here, $$\ell$$ is the hyperbolic length, $$\tilde{\gamma}$$ is the unique simple closed geodesic homotopic to $$\gamma$$, and, to be clear, $$\ell(\gamma|_{\Sigma\backslash C})$$ refers to the length of the restriction of $$\gamma$$ to $$\Sigma\backslash C$$ (this is a union of disconnected curves). Here $$d$$ should depend on the hyperbolic metric and the choice of $$C$$, but not the homotopy class of the curve.

The reason I add the "$$+d$$" is that one could cook up the following example: take a curve that enters $$C$$ once, and then replace the region that enters $$C$$ with a straight line connecting the boundary points at which the curve crosses. We then get a closed curve homotopic to $$\tilde{\gamma}$$, and the length is at most $$\ell(\gamma|_{\Sigma\backslash C})+d$$, where $$d=\ell(\partial C)$$.

It seems true for just about any example I draw, but any proof I attempt to write down becomes terribly buggy and complicated.

There is no such constant $$d$$. The "reason" is that geodesic laminations exist, and are approximated by simple closed curves.

Thought process (which can be ignored):

Suppose that $$\tau$$ is a train track in $$\Sigma$$. Let $$R$$ be a rectangle embedded in $$\Sigma - \tau$$ with one side running along $$\partial \Sigma$$ and one side running along a branch $$b$$ of $$\tau$$. If $$\gamma$$ is carried by $$\tau$$ then we can use $$R$$ to perform a homotopy of $$\gamma$$; we take all of the arcs of $$\gamma$$ running along $$b$$ and isotope a subsegment of each, along $$R$$ into $$C$$. We call the sides of $$R$$ in $$\partial \Sigma$$ and in $$b$$ "horizontal" and we call the other two sides "vertical".

Let $$\gamma'$$ be the result of the homotopy. After the homotopy we have added many copies of the vertical sides (to the length of $$\gamma' - C$$) and subtracted the same number of many copies of the segment along $$b$$.

So if we can find a surface $$\Sigma$$ equipped with a track $$\tau$$ where $$R$$ has short verticals and long horizontals, we are done.

The resulting proof:

Suppose that $$\Sigma'$$ is a closed surface with a very short geodesic curve $$\alpha$$. Let $$\beta$$ and $$\beta'$$ be distinct, disjoint, simple geodesic loops that cross $$\alpha$$ exactly once. It will be useful to arrange for $$\beta$$ to be separating, and $$\beta'$$ to be non-separating.

Let $$C_\alpha$$ be a collar of $$\alpha$$ where $$\partial C_\alpha$$ is much smaller than the "height" of $$C_\alpha$$. Since $$\alpha$$ is short, the curves $$\beta$$ and $$\beta'$$ fellow travel, very closely, inside of $$C_\alpha$$. By Dehn twisting about $$\beta'$$ we can now produce a family of curves $$\gamma_n$$ that are (a) disjoint from $$\beta$$ and (b) cross $$\alpha$$ exactly $$n$$ times. We produce $$\Sigma$$ by cutting along $$\beta$$ and throwing away the component that does not contain the $$\gamma_n$$.

Let $$C_\beta$$ be the given collar of $$\partial \Sigma = \beta$$. We now perform a homotopy of $$\gamma_n$$ (supported in the remains of $$C_\alpha$$) so that the $$n$$ long arcs of $$\gamma_n \cap C_\alpha$$ lie inside of $$C_\beta$$. The length of $$\gamma_n - C$$ is less than that of $$\gamma_n$$ by (approximately) $$n$$ times the difference between the height of $$C_\alpha$$ and the length of $$\partial C_\alpha$$.