Discrete subgroup

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In mathematics , discrete subsets of topological groups play an important role in topology , differential geometry, and the theory of Lie groups .


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 .


  • is a discrete subgroup
  • is a discrete subgroup
  • is not a discrete subgroup
  • is a discrete subgroup


A discrete subgroup of a Hausdorff topological group is always closed.


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.


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