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

  • Pablo Lessa

    Universidad de la República, Montevideo, Uruguay
  • Ernesto Garcia

    Universidad de la República, Montevideo, Uruguay
On the discreteness of states accessible via right-angled paths in hyperbolic space cover

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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 along the geodesic in direction of the first vector, or rotate it in place a right angle. We characterize the values of 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 (see [Gil2] and the references therein) for a particular one parameter family of two-generator subgroups of . 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.

Cite this article

Pablo Lessa, Ernesto Garcia, On the discreteness of states accessible via right-angled paths in hyperbolic space. Enseign. Math. 66 (2020), no. 3/4, pp. 383–407

DOI 10.4171/LEM/66-3/4-4