Details of Grant 

EPSRC Reference:	GR/L48331/01
Title:	PLEATING COORDINATES FOR KLEINIAN GROUPS
Principal Investigator:	Series, Professor C
Other Investigators:
Researcher Co-Investigators:
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Department:	Mathematics
Organisation:	University of Warwick
Scheme:	Standard Research (Pre-FEC)
Starts:	16 April 1997	Ends:	15 April 2000	Value (£):	2,700
EPSRC Research Topic Classifications:	Algebra & Geometry
EPSRC Industrial Sector Classifications:	No relevance to Underpinning Sectors
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Summary on Grant Application Form
The topic is the continuation of the pleating coordinates programme to understand deformation spaces of Kleinian groups. The deformation space is partitioned into subsets, called pleating varieties, on which the convex hull boundary of the group is pleated along a specific projective measure class of geodesic laminations. We focus our attention on those pleating varieties for which the bending lamination is a union of disjoint simple closed curves. On such a pleating variety the traces of the group elements corresponding to the geodesics in the bending lamination are real. The converse is not necessarily true, nevertheless expressing the trace as a rational function of an appropriately chosen set of parameters, we show that the pleating variety is a connected, non-empty and non-singular branch of the real trace variety. Further, we identify explicitly exactly which branch by studying the asymptotic behaviour of the trace function. Particular features of current investigation are the formulae for top terms of trace polynomials in relation to carefully chosen train track coordinates for the bending lamination, the way in which the pleating varieties meet the sublocus of Fuchsian groups and an analysis of lower dimensional subsets in which the lengths of certain of the curves in the bending lamination remain constant. Also of interest are the boundary of the deformation space and the extension of the pleating varieties outside the space.
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Organisation Website:	http://www.warwick.ac.uk