12 The Spectral Geometry of Riemann Surfaces 235- [2]. This paper shows how to construct Riemann surface with large first
eigenvalue of arbitrary genus. It also contains growth estimates for eigen-
values in cusps as well as other generally useful techniques involving com-
pactification and the behavior of eigenvalues. - [7]. Contains a number of versions of the Ahlfors–Schwarz lemma, includ-
ing the version needed in [2]. - [3]. Features graph-theoretic techniques for building Belyi surfaces with
various nice properties. - [8]. This contains the theorem relating Cheeger constants of graphs to the
behavior of the first eigenvalue under coverings. (Too old to be available
at my web site.) - [9]. A survey article discussing the construction of building surfaces from
three-regular graphs.
In addition to these papers, I would like to mention the M.Sc. thesis of
my student Dan Mangoubi, “Riemann Surfaces and three-Regular Graphs,”
available from my web site. In addition to giving a good overview of the sub-
ject, it contains interesting quantitative results elaborating on the qualitative
theory of [6]. There are also two papers in preparation: my paper with An-
drzej Zuk on Cheeger constants of graphs and surfaces, and my paper with
Mikhail Monastyrsky, which generalizes the theory from three-regular graphs
tok-regular graphs.
References
- R.Brooks,E.Makover,ERA-AMS 5 76–81 (1999)
- R.Brooks,E.Makover,J.d’Anal. 83 243–258 (2001)
- R.Brooks,E.Makover,Sodinet.al(eds.),Entire functions in modern analysis,
IMCP 15 37–46 (2002) - R. Brooks, E. Makover, Preprint, Department of Mathematics, Technion (1997)
- R. Brooks, Comm. Math. Helv. 56 581–596 (1981)
- R. Brooks, Comm. Math. Helv. 74 156–170 (1999)
- R. Brooks, Brooks and Sodin (eds.), Lectures in Memory of Lars Ahlfors, IMCP
14 , 31–39 - R.Brooks,J.Diff.Geom. 23 97–107 (1986)
- R. Brooks, Picardello and Woess (eds.),Random walks and discrete potential
theory, (Camb. Univ. Press, 1999), pp. 85–103