Homogeneous Riemann surfaces
Ara Basmajian
City University of New York (USA)
Abstract:
There are various contexts for which it is natural to study spaces that look the same from any point of the space (that is, homogeneous spaces). For example, the simply connected Riemann surfaces, namely the Riemann sphere, complex plane, and unit disc are conformally homogeneous Riemann surfaces. In fact they are the only ones. On the other hand, given any surface it is not difficult to cook up a diffeomorphism between any two points of the surface. Hence one needs a notion that is not as strong as conformality and not as weak as differentiability. The key observation is that while smooth maps can distort infinitesimal circles to ellipses with unbounded eccentricity (the ratio of the major to the minor axis can be arbitrarily large), conformal maps do not distort infinitesimal circles at all. This leads to the notion of a homeomorphism being 
\[K\]
-quasiconformal (has eccentricity bounded by 
\[K\]
). Conformal homeomorphisms are 
\[1\]
-quasiconformal.
A Riemann Surface
\[X\]
is said to be
\[K\]
-quasiconformally homogeneous if for any two points
\[x\]
and
\[y\]
on it, there exists a
\[K\]
-quasiconformal self-mapping taking
\[x\]
to
\[y\]
. If such a
\[K\]
exists we say that
\[X\]
is a
\[QCH\]
surface. It is not difficult to show that the regular cover of a closed Riemann surface is
\[QCH\]
, and hence the quasiconformal deformation of such a regular cover is also
\[QCH\]
. This leads us to the question to what extent being
\[qc\]
equivalent to a regular cover characterizes
\[QCH\]
surfaces. Bonfert-Taylor, Canary, Souto, and Taylor showed that there are
\[QCH\]
Riemann surfaces that are not
\[qc\]
-equivalent to the regular cover of a closed surface. On the other hand, in joint work with Nicholas Vlamis we show that all
\[QCH\]
ladder surfaces are
\[qc\]
-equivalent to the regular cover of a closed surface. After an introduction to the basics we will discuss the proof of this theorem which involves the hyperbolic geometry of the surface.