Proceedings of Symposia in Pure Mathematics
Volume 0, XXXX
Introduction to Spectral Theory on Hyperbolic Surfaces
David Borthwick
Contents
1. Hyperbolic geometry
2. Fuchsian groups and hyperbolic surfaces
3. Spectrum and resolvent
4. Spectral theory: finite-area case
5. Spectral theory: infinite-area case
6. Selberg trace formula
7. Arithmetic surfaces
References
1. Hyperbolic geometry
In complex analysis, we learn that the upper half-plane H = {Im z 0} has a
large group of conformal automorphisms, consisting of obius transformations of
the form
(1.1) γ : z
az + b
cz + d
,
where a, b, c, d R and ad bc 0. The Schwarz Lemma implies that all au-
tomorphisms of H are of this type. Since the transformation is unchanged by a
simultaneous rescaling of a, b, c, d, the conformal automorphism group of H is iden-
tified with the matrix group
PSL(2, R) := SL(2, R)/{±I}.
Under the PSL(2, R) action, H has an invariant metric,
dsH
2
=
dx2
+
dy2
y2
,
often called the Poincar´ e metric. To see the invariance, it is convenient to switch
to the complex notation, where
dsH
2
=
|dz|2
(Im z)2
.
2000 Mathematics Subject Classification. Primary 58J50, 35P25; Secondary 47A40.
Supported in part by NSF grant DMS-0901937.
c XXXX American Mathematical Society
1
Proceedings of Symposia in Pure Mathematics
Volume 84 , 2012
c 2012 American Mathematical Society
3
http://dx.doi.org/10.1090/pspum/084/1347
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