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

Ernesto Garcia; Pablo Lessa

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arXiv · 1911.06853

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

Ernesto Garcia·Pablo Lessa

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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.

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Ernesto Garcia, Pablo Lessa. 2019-11-15. On the discretness of states accessible via right-angled paths in hyperbolic space. https://arxiv.org/abs/1911.06853

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