Collar lemma
= Collar lemma
{title2=$\sinh w\,\sinh(\ell/2)=1$}
A <simple closed curve> that is a <geodesic> of length $\ell$ on a <hyperbolic surface> has an embedded collar of half-width $w$ with $\sinh w\,\sinh(\ell/2)=1$. In circumference-one coordinates its metric is $dr^2+\ell^2\cosh^2r\,dt^2$, and its <conformal modulus of an annulus> is $(2/\ell)\arctan(\operatorname{csch}(\ell/2))\sim\pi/\ell$.