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

TL;DR

This paper investigates the conditions under which the set of orthonormal frames accessible via specific right-angled paths in hyperbolic space is discrete, linking geometric control problems to subgroup discreteness in hyperbolic isometry groups.

Contribution

It characterizes the values of step size r for which the accessible frames form a discrete set, solving a specific discreteness problem in hyperbolic geometry.

Findings

Discreteness conditions for hyperbolic plane cases.

Discreteness conditions for three-dimensional hyperbolic space.

Connection between control paths and subgroup discreteness in PSL_2(ℝ).

Abstract

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.