Simple geodesics in closed hyperbolic manifolds

Given any closed hyperbolic n-manifold M ($n\ge 3$), we show that for a fixed $κ>\frac{3}{n-2}$, a random closed geodesic of length close to r has the collar of width $r^{-κ}$ when r is large enough. In particular, most closed geodesics of length close to r are simple. This theorem positively answer