Restricted Chebotarev Theorems for $\mathrm{SL}(2 , \mathbb{Z})$ and related groups

The “Chebotarev Prime Geodesic Theorem” for primitive conjugacy classes in $\mathrm{SL}(2 , \mathbb{Z})$ (which correspond to closed geodesics on the quotient) is well known and can be proved using the Selberg Trace Formula. We examine local versions of this Theorem where the closed geodesics are restricted in various ways and in particular to being simple. Congruence and arithmetic features intervene decisively as well as transitivity properties of the action of associated mapping class groups.