2018/11/16 by David Aulicino · 1 voice
Mathematics · Physics and Astronomy · #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems #math.GT
paper · pdf · doi:10.1216/rmj.2020.50.9
arxiv published 2018/11/16 · openalex created_date 2018/11/29 · arxiv updated 2019/08/30 · openalex publication_date 2020/02/01 · openalex updated_date 2026/07/28
We borrow a classical construction from the study of rational billiards in dynamical systems known as the “unfolding construction” and show that it can be used to study the automorphism group of a Platonic surface. More precisely, the monodromy group, or deck group in this case, associated to the cover of a regular polygon or double polygon by the unfolded Platonic surface yields a normal subgroup of the rotation group of the Platonic surface. The quotient of this rotation group by the normal subgroup is always a cyclic group, where explicit bounds on the order of the cyclic group can be given entirely in terms of the Schläfli symbol of the Platonic surface. As a consequence, we provide a new derivation of the rotation groups of the dodecahedron and the Bolza surface.