Journal of Differential Geometry

Random Construction of Riemann Surfaces

Robert Brooks and Eran Makover

Abstract

We develop a new approach for the study of “typical” Riemann surfaces with high genus. The method that we use is the construction of random Riemann surfaces from oriented cubic graphs. This construction enables us to get a control over the global geometry properties of compact Riemann surfaces. We use the theory of random regular graphs to show that almost all such surfaces have large first eigenvalues and large Cheeger constants. Moreover a closer analysis of the probability space of oriented cubic graphs shows that on a typical surface there is a large embedded hyperbolic ball.

Article information

Source
J. Differential Geom., Volume 68, Number 1 (2004), 121-157.

Dates
First available in Project Euclid: 8 December 2004

Permanent link to this document
https://projecteuclid.org/euclid.jdg/1102536712

Digital Object Identifier
doi:10.4310/jdg/1102536712

Mathematical Reviews number (MathSciNet)
MR2152911

Zentralblatt MATH identifier
1095.30037

Citation

Brooks, Robert; Makover, Eran. Random Construction of Riemann Surfaces. J. Differential Geom. 68 (2004), no. 1, 121--157. doi:10.4310/jdg/1102536712. https://projecteuclid.org/euclid.jdg/1102536712


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