On the discretness of states accessible via right-angled paths in hyperbolic space

We consider the control problem where, given an orthonormal tangent frame in the hyperbolic plane or three dimensional hyperbolic space, one is allowed to transport the frame a fixed distance $r > 0$ along the geodesic in direction of the first vector, or rotate it in place a right angle. We characterize the values of $r > 0$ for which the set of orthonormal frames accessible using these transformations is discrete. In the hyperbolic plane this is equivalent to solving the discreteness problem for a particular one parameter family of two-generator subgroups of $PSL_2(\mathbb{R})$. In the three dimensional case we solve this problem for a particular one parameter family of subgroups of the isometry group which have four generators.

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