Definition
Kleinian group
A discrete subgroup of PSL(2,C), acting on hyperbolic three-space and its boundary.
A Kleinian group is a discrete subgroup , where the ambient group has its usual real Lie-group topology. Through its standard action, it is a discrete group of orientation-preserving isometries of hyperbolic three-space.
Scope
This convention includes finite and elementary groups. Finite covolume, torsion-freeness, and arithmeticity are additional conditions, not part of the definition.
Examples
The group generated by is Kleinian. It has infinite hyperbolic covolume. Bianchi groups supply finite-covolume examples with torsion.
Two actions
The action on hyperbolic three-space extends to the Riemann sphere at infinity. Statements about fixed points must specify whether they concern the interior or this boundary.
References
- F. W. Gehring, C. Maclachlan, G. J. Martin, and A. W. Reid, Arithmeticity, discreteness and volume, Transactions of the AMS 349 (1997). Author-hosted paper, §§2–4.