2024/09/12 by Sasaki, Dounnu · 1 citation
#20F34 #30F35 #FOS: Mathematics #Geometric Topology (math.GT)
paper · doi:10.48550/arxiv.2409.08109
Given a hyperbolic surface Σ of genus g with r cusps, Mirzakhani proved that the number of closed geodesics of length at most L and of a given type is asymptotic to cL6g-6+2r for some c>0. Since a closed geodesic corresponds to a conjugacy class of the fundamental group π1(Σ), we extend this to the counting problem of conjugacy classes of finitely generated subgroups of π1(Σ). Using `half the sum of the lengths of the boundaries of the convex core of a subgroup' instead of the length of a closed geodesic, we prove that the number of such conjugacy classes is similarly asymptotic to cL6g-6+2r for some c>0. As a special case, these conjugacy classes can be interpreted as subsurfaces of Σ via their convex cores, and the result can be viewed as counting subsurfaces of a given type. Furthermore, we see that the above length measurement for subgroups is `natural' within the framework of subset currents, which serve as a completion of weighted conjugacy classes of finitely generated subgroups of π1(Σ).