Modular symbols for Teichmüller curves
type: Item
instance of: scholarly article
title: Modular symbols for Teichmüller curves
zbMATH Open document ID: 1477.37058
author: Curtis T. McMullen
published in: Journal für die Reine und Angewandte Mathematik
publication date: 2021-08-03T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 37F34
Mathematics Subject Classification ID: 30F30
Mathematics Subject Classification ID: 30F60
Mathematics Subject Classification ID: 37C83
Mathematics Subject Classification ID: 37A30
zbMATH DE Number: 7377469
zbMATH Keywords: nonabelian modular symbols
zbMATH Keywords: hyperbolic Riemann surface
zbMATH Keywords: polygonal billiards
zbMATH Keywords: holomorphic 1-forms
reviewed by: Shengjin Huo
MaRDI profile type: Publication
full work available at URL: crelle-2021-0019
OpenAlex ID: W3159981860
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DOI: 10.1515/CRELLE-2021-0019
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description: scientific article; zbMATH DE number 7377469