Limits of canonical forms on towers of Riemann surfaces

TY - JOUR

T1 - Limits of canonical forms on towers of Riemann surfaces

AU - Baik, Hyungryul

AU - Shokrieh, Farbod

AU - Wu, Chenxi

PY - 2020/7/1

Y1 - 2020/7/1

N2 - We prove a generalized version of Kazhdan's theorem for canonical forms on Riemann surfaces. In the classical version, one starts with an ascending sequence ÂSn S° of finite Galois covers of a hyperbolic Riemann surface S, converging to the universal cover. The theorem states that the sequence of forms on S inherited from the canonical forms on Sn's converges uniformly to (a multiple of) the hyperbolic form. We prove a generalized version of this theorem, where the universal cover is replaced with any infinite Galois cover. Along the way, we also prove a Gauss-Bonnet-type theorem in the context of arbitrary infinite Galois covers.

AB - We prove a generalized version of Kazhdan's theorem for canonical forms on Riemann surfaces. In the classical version, one starts with an ascending sequence ÂSn S° of finite Galois covers of a hyperbolic Riemann surface S, converging to the universal cover. The theorem states that the sequence of forms on S inherited from the canonical forms on Sn's converges uniformly to (a multiple of) the hyperbolic form. We prove a generalized version of this theorem, where the universal cover is replaced with any infinite Galois cover. Along the way, we also prove a Gauss-Bonnet-type theorem in the context of arbitrary infinite Galois covers.

UR - http://www.scopus.com/inward/record.url?scp=85065601746&partnerID=8YFLogxK

U2 - 10.1515/crelle-2019-0007

DO - 10.1515/crelle-2019-0007

M3 - Journal article

AN - SCOPUS:85065601746

SN - 0075-4102

JO - Journal fur die Reine und Angewandte Mathematik

JF - Journal fur die Reine und Angewandte Mathematik

ER -