Recent zbMATH articles in MSC 30F35https://www.zbmath.org/atom/cc/30F352022-05-16T20:40:13.078697ZWerkzeugOn the discreteness of states accessible via right-angled paths in hyperbolic spacehttps://www.zbmath.org/1483.300832022-05-16T20:40:13.078697Z"Lessa, Pablo"https://www.zbmath.org/authors/?q=ai:lessa.pablo"Garcia, Ernesto"https://www.zbmath.org/authors/?q=ai:garcia.ernestoSummary: 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 (see [\textit{J.Gilman}, Geom. Dedicata 201, 139--154 (2019; Zbl 1421.30056)] and the references therein) for a particular one parameter family of two-generator subgroups of \(\mathrm{PSL}_2(\mathbb{R})\). 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.