Modular symbols for Teichmüller curves
The author mainly investigates the following four related phenomena: the behavior of long periodic trajectories for billiards in a regular polygon; the distribution of the closed geodesics on a singular flat surface \((X, |\omega|)\), when \(SL(X, \omega)\) is a lattice; the different ways of describing a Teichmüller curve by purely topological data, and the closure of the projective monodromy group for such a curve. In all four cases, a countable compact set homeomorphic to the ordinal \(\omega^{\omega}+1\) emerges. The author shows that these sets are all reflections of the space of nonabelian modular symbols \(\mathcal{S}(V)\), which is itself homeomorphic to \(\omega^{\omega}+1\). This new structure is a natural enhancement of the classical modular symbols for \(\text{SL}_{2}(\mathbb{Z}).\) Applying the new structure, he obtains new results on polygonal billiards and holomorphic \(1\)-forms. In particular, his results show that the scarring behavior of periodic trajectories for billiards in a regular polygon is governed by a countable set of measures homeomorphic to \(\omega^{\omega}+1, \) which completes the description of the closure of the ergodic invariant measures for billiards in a regular polygon.
scientific article; zbMATH DE number 7377469
scientific article