Complex Kleinian Groups (progress In Mathematics)
by José Seade /
2012 / English / PDF
3.2 MB Download
This monograph lays down the foundations of the theory of complex
Kleinian groups, a newly born area of mathematics whose origin
traces back to the work of Riemann, Poincaré, Picard and many
others. Kleinian groups are, classically, discrete groups of
conformal automorphisms of the Riemann sphere, and these can be
regarded too as being groups of holomorphic automorphisms of the
complex projective line CP1. When going into higher dimensions,
there is a dichotomy: Should we look at conformal automorphisms of
the n-sphere?, or should we look at holomorphic automorphisms of
higher dimensional complex projective spaces? These two theories
are different in higher dimensions. In the first case we are
talking about groups of isometries of real hyperbolic spaces, an
area of mathematics with a long-standing tradition. In the second
case we are talking about an area of mathematics that still is in
its childhood, and this is the focus of study in this monograph.
This brings together several important areas of mathematics, as for
instance classical Kleinian group actions, complex hyperbolic
geometry, chrystallographic groups and the uniformization problem
for complex manifolds.
This monograph lays down the foundations of the theory of complex
Kleinian groups, a newly born area of mathematics whose origin
traces back to the work of Riemann, Poincaré, Picard and many
others. Kleinian groups are, classically, discrete groups of
conformal automorphisms of the Riemann sphere, and these can be
regarded too as being groups of holomorphic automorphisms of the
complex projective line CP1. When going into higher dimensions,
there is a dichotomy: Should we look at conformal automorphisms of
the n-sphere?, or should we look at holomorphic automorphisms of
higher dimensional complex projective spaces? These two theories
are different in higher dimensions. In the first case we are
talking about groups of isometries of real hyperbolic spaces, an
area of mathematics with a long-standing tradition. In the second
case we are talking about an area of mathematics that still is in
its childhood, and this is the focus of study in this monograph.
This brings together several important areas of mathematics, as for
instance classical Kleinian group actions, complex hyperbolic
geometry, chrystallographic groups and the uniformization problem
for complex manifolds.
