179 research outputs found
Random Construction of Riemann Surfaces
In this paper, we address the following question: What does a typical compact
Riemann surface of large genus look like geometrically? We do so by
constructing compact Riemann surfaces from oriented 3-regular graphs. The set
for such Riemann surfaces is dense in the space of all compact Riemann
surfaces, namely Belyi surfaces. And in this construction we can control the
geometry of the compact Riemann surface by the geometry of the graph. We show
that almost all such surfaces have large first eigenvalue and large Cheeger
constant
Some counterexamples in surface homology
We present four counterexamples in surface homology. The first example shows
that even if the loops inducing a homology basis intersect each other at most
once, they still may separate the surface into two parts. The other three
examples show some difficulties in working with minimal homology bases.Comment: 14 pages, 8 figure
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