Quasi-isometric embeddings for shrinking maps from surfaces into the moduli space

2025/11/12 by Zhang, Yibo
#30 F 45 #37 F 34 #51 F 30 #57 K 20 #FOS: Mathematics #Geometric Topology (math.GT) #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.2511.09317

Abstract

We investigate shrinking maps from a cusped hyperbolic surface into the moduli space of closed Riemann surfaces. For such a map and its lift to the Teichmüller space, we consider whether they are quasi-isometric embeddings with respect to natural metrics like the Teichmüller distance and the intrinsic distance. Under a mild condition, we prove that these properties are characterised solely by the map's monodromy. These characterisations apply, in particular, to holomorphic maps.

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