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Spaces of Kleinian Groups and Hyperbolic 3-Manifolds

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21st July 2003 to 15th August 2003
Caroline Series University of Warwick
Yair Minsky [University of New York], Yale University
Makoto Sakuma Osaka University


Programme theme

Kleinian groups stand at the meeting point of several different parts of mathematics. Classically, they arose as the monodromy groups of Schwarzian equations, in modern terminology projective structures, on Riemann surfaces.The action by Möbius maps on the Riemann sphere provides further intimate links not only with Riemann surfaces, but also complex dynamics and fractals, while Thurston's revolutionary insights have made hyperbolic 3-manifolds, quotients of the isometric action on hyperbolic 3-space, central to 3-dimensional topology.

The juxtaposition of Thurston's work with Teichmüller theory makes it natural to study spaces of Kleinian groups. From the viewpoint of complex dynamics, a space of Kleinian groups is a close analogue of the Mandelbrot set; Bers' simultaneous uniformisation theorem reveals these spaces as an extension of Teichmüller theory; while from the 3-dimensional viewpoint they become deformation spaces of hyperbolic 3-manifolds. Such ideas will be the main focus of this meeting.Each approach contributes its individual flavour, and the aim of this programme is to bring together the diverse threads.

Specific topics to be addressed include relationships between the analytic, combinatorial and geometric structure of hyperbolic 3-manifolds; topology of deformation spaces and the arrangement of their components; classification of hyperbolic 3-manifolds by asymptotic invariants; complex projective structures; convex hull boundaries; cone manifolds, orbifolds and knot groups; the combinatorial structure of Teichmüller spaces, mapping class groups,and spaces of curves on surfaces; and the challenge of extending recent advances from once punctured tori to higher genus.

Throughout the development of this subject, exploratory computer graphics have consistently played an important role. A particular focus of this programme will be to foster such experimentation. In particular, some remarkable recent pictures of parameter spaces indicate the presence of unexpected new phenomena. We anticipate that intensive interactions between authors of graphics programs and leading mathematicians will make deep inroads into such problems and be formative for new phases of research.

The high point of the meeting will be a conference bringing together additional participants to discuss ongoing developments in this rapidly developing area.

Final Scientific Report: 
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons