Discrete subgroup
In mathematics , discrete subsets of topological groups play an important role in topology , differential geometry, and the theory of Lie groups .
definition
Be a topological group. A subgroup is called discrete if the induced subspace topology is the discrete topology , i.e. all elements are isolated : there are no other elements of in a sufficiently small neighborhood of any element .
A representation of an (abstract) group is called discrete if the image is a discrete subgroup of .
Examples
- is a discrete subgroup
- is a discrete subgroup
- is not a discrete subgroup
- is a discrete subgroup
properties
A discrete subgroup of a Hausdorff topological group is always closed.
Grid
Let be a locally compact -compact topological group, the projection and the hair measure (which is unique except for a constant factor) . For a discrete subgroup the Hair measure creates a well-defined measure on as follows: for all sets with we define .
A lattice is a discrete subgroup for which there is a fundamental domain of finite volume, or equivalent: for which the quotient space has finite volume (with regard to the hair measure).
The grid is called uniform or co-compact if it is compact.
A grid is reducible when as a direct product can be disassembled so that it lattice are, for a subgroup of finite index in , and irreducible , if there is not such a decomposition.
literature
- MS Raghunathan: Discrete subgroups of Lie groups . Results of mathematics and its border areas. tape 68 . Springer, New York, Heidelberg 1972.
- GA Margulis : Discrete subgroups of semisimple lie groups , results of mathematics and their border areas (3), 17. Springer , Berlin, 1991. ISBN 3-540-12179-X
Web links
- Venkataramana: Lattices in Lie groups