Volume entropy and the Gromov boundary of flat surfaces

2011/01/10 by Klaus Dankwart, Dankwart, Klaus
Mathematics · #30F30 #30F60 #53C99 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #Metric Geometry (math.MG)

paper · pdf · doi:10.48550/arxiv.1101.1795

openalex publication_date 2011/01/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the volume entropy of closed flat surfaces of genus g≥ 2 and area 1. We show that a sequence of flat surfaces diverges in the moduli space if and only if the volume entropy converges to infinity. Equivalently the Hausdorff dimension of the Gromov boundary of the isometric universal cover tends to infinity. Moreover, we estimate the entropy of a locally isometric branched covering of a flat surface by the entropy of base surface and the geometry of the covering map.

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