Counting lariats
On a hyperbolic surface a lariat is a bi infinite simple geodesic with both ends at a common cusp. The reduced length of a lariat is defined by choosing a horocycle for truncation. We discuss adapting Mirzakhani’s method from “Growth of the number of simple closed geodesics …” to count by length the lariats at a given cusp. The method combines counting and estimating the integral lattice points in $\mathcal{MGL}$, the space of measured geodesic laminations; Masur’s ergodicity for the mapping class group acting on $\mathcal{MGL}$ and integration over the moduli space to evaluate constants. The overall result is that lariats counted by length have the same growth rate as simple closed geodesics and as in Mirzakhani’s result, the leading constant in the count is given by characteristic classes on the moduli space. The original method is adapted by counting cosets in the mapping class group. The approach is applied to count pants decompositions with controlled ratios for lengths. We discuss applying the method to counting Markoff forms, special binary indefinite quadratic forms.