A Note on Poincaré’s Polyhedron Theorem in Complex Hyperbolic Space

2018 ◽  
pp. 351-361
Author(s):  
Li-Jie Sun
Keyword(s):  
Hyperbolic Space ◽  
Complex Hyperbolic Space

SHIMIZU’S LEMMA FOR COMPLEX HYPERBOLIC SPACE

1992 ◽  
Vol 03 (02) ◽  
pp. 291-308 ◽  
Author(s):  
JOHN R. PARKER
Keyword(s):  
Hyperbolic Space ◽  
Hyperbolic Plane ◽  
Discrete Group ◽  
Complex Hyperbolic Space ◽  
Necessary Condition ◽  
Higher Dimensional ◽  
Group Of Isometries ◽  
Vertical Translation

Shimizu’s lemma gives a necessary condition for a discrete group of isometries of the hyperbolic plane containing a parabolic map to be discrete. Viewing the hyperbolic plane as complex hyperbolic 1-space we generalise Shimizu’s lemma to higher dimensional complex hyperbolic space In particular we give a version of Shimizu’s lemma for subgroups of PU (n, 1) containing a vertical translation Partial generalisation to groups containing either an ellipto-parabolic map or non-vertical translations are also given together with examples that show full generalisation is not possible in these cases

scholarly journals Incidence Axioms for the Boundary at Infinity of Complex Hyperbolic Spaces

2015 ◽  
Vol 3 (1) ◽  
Author(s):  
Sergei Buyalo ◽  
Viktor Schroeder
Keyword(s):  
Hyperbolic Space ◽  
Complex Hyperbolic Space ◽  
Hyperbolic Spaces ◽  
Complex Hyperbolic Spaces

Abstract We characterize the boundary at infinity of a complex hyperbolic space as a compact Ptolemy space that satisfies four incidence axioms.

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