Are bounded geodesics in the modular surface closed?

Question

Let $M=\mathbb{H}/SL(2,\mathbb{Z})$ be the modular surface (which is noncompact but finite volume with the volume induced by the constant negative curvature metric inherited from $\mathbb{H}$). Any closed geodesic on $M$ is obviously bounded and there are uncountably many unbounded geodesics. The question is then: is there a bounded geodesic which is not closed?

Answer 1

Yes, because there are uncountably many unbounded geodesics and only countably many closed geodesics. The closed geodesics correspond one-to-one with subset of the set of conjugacy classes of infinite cyclic subgroups, and there are only countably many of the latter. The endpoints of bounded rays, when lifted to $\mathbb{H}^2$, are precisely the real numbers with bounded continued fraction expansions, of which there are uncountably many. There are uncountably many pairs of such points, leading to uncountably many geodesics in $\mathbb{H}^2$ whose image in $M$ is bounded.