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The shape of the filling-systole subspace in surface moduli space and critical points of the systole function

Yue Gao

Algebraic & Geometric Topology 24 (2024) 2011–2038
Abstract

We study the space Xg g consisting of surfaces with filling systoles and its subset, critical points of the systole function. In the first part we obtain a surface with Teichmüller distance 1 5 log log g to Xg, and in the second and third parts prove that most points in g have Teichmüller distance 1 5 log log g and Weil–Petersson distance 0.6521(log g 7 log log g) to Xg. So the radius-r neighborhood of Xg cannot cover the thick part of g for any fixed r > 0. In the last two parts, we get critical points with small and large (comparable to the diameter of the thick part of g) distances.

Keywords
filling systole subspace, sparseness, random surfaces, Weil–Petersson metric, critical points of the systole function
Mathematical Subject Classification
Primary: 30F45, 30F60, 57K20
References
Publication
Received: 7 February 2022
Revised: 17 January 2023
Accepted: 27 April 2023
Published: 16 July 2024
Authors
Yue Gao
School of Mathematics and Statistics
Anhui Normal University
Wuhu
China

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