Rigidity in Complex Projective Space.

Discrete Subgroups of Lie Groups;Rigidity;Hilbert Metric;Convexity;Mathematics;Science;Mathematics

Zimmer, Andrew M.Prasad, Gopal ;

University of Michigan

关键词: Discrete Subgroups of Lie Groups;    Rigidity;    Hilbert Metric;    Convexity;    Mathematics;    Science;    Mathematics;   

Others  :  https://deepblue.lib.umich.edu/bitstream/handle/2027.42/108912/aazimmer_1.pdf?sequence=1&isAllowed=y

瑞士|英语

来源: The Illinois Digital Environment for Access to Learning and Scholarship

【 摘 要 】

An open convex set in real projective space is called divisible if there exists a discrete group of projective automorphisms which acts co-compactly. There are many examples of such sets and a theorem of Benoist implies that many of these examples are strictly convex, have C1 boundary, and have word hyperbolic dividing group. In this thesis we study notions of convexity in complex and quaternionic projective space and show that the only divisible ``convex;;;; sets with C1 boundary are the projective balls. A key tool in these arguments is an analogue of the classical Hilbert metric. These new metrics prove to be useful in the complex and quaternionic setting but have the downfall that they are rarely geodesic. In fact we will prove that these metrics are geodesic if and only if the underlying set is a projective ball. Moreover, when the underlying set is a projective ball these metrics provide a model of complex or quaternionic hyperbolic space.