Benjamini-Schramm Convergence of Normalized Characteristic Numbers of Riemannian Manifolds

by Daniel Luckhardt
Doctoral thesis
Date of Examination:2018-06-05
Date of issue:2020-04-30
Advisor:Prof. Dr. Thomas Schick
Referee:Prof. Dr. Thomas Schick
Referee:Prof. Dr. Ralf Meyer
Referee:Prof. Dr. Stephan Huckemann
Referee:Prof. Dr. Russell Luke
Referee:Prof. Dr. Viktor Pidstrygach
Referee:Prof. Dr. Ingo Witt
Persistent Address: http://dx.doi.org/10.53846/goediss-7955

 

 

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Abstract

English

We study a weak form of Gromov-Hausdorff convergence of Riemannian manifolds, also known as Benjamini-Schramm convergence. This concept is also applicable to other areas and has widely been studied in the context of graphs. The main result is the continuity of characteristic numbers normalized by the volume with respect to the Benjamini-Schramm topology on the class of Riemannian manifolds with a uniform lower bound on injectivity radius and Ricci curvature. An immediate consequence is a comparison theorem that gives for any characteristic number a linear bound in terms of the volume on the entire class of manifolds mentioned. We give another interpretation of the result showing that characteristic numbers can be reconstructed with some accuracy from local random information.
Keywords: Benjamini-Schramm convergence; Riemannian manifold; characteristic number; Gromov-Hausdorff convergence; metric measure spaces
 
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