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 . ${\displaystyle G}$ ${\displaystyle \Gamma }$ ${\displaystyle \gamma \in \Gamma }$ ${\displaystyle \Gamma }$

A representation of an (abstract) group is called discrete if the image is a discrete subgroup of . ${\displaystyle \rho \colon \Gamma \to GL(n,\mathbb {C} )}$ ${\displaystyle \Gamma }$ ${\displaystyle \rho (\Gamma )}$ ${\displaystyle GL(n,\mathbb {C} )}$

Examples

• ${\displaystyle \mathbb {Z} \subset \mathbb {R} }$ is a discrete subgroup
• ${\displaystyle \mathbb {Z} \subset \mathbb {C} }$ is a discrete subgroup
• ${\displaystyle \mathbb {Q} \subset \mathbb {C} }$ is not a discrete subgroup
• ${\displaystyle GL(n,\mathbb {Z} )\subset GL(n,\mathbb {R} )}$ 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 . ${\displaystyle G}$ ${\displaystyle \sigma }$ ${\displaystyle \pi \colon G\rightarrow \Gamma \backslash G}$ ${\displaystyle \mu }$ ${\displaystyle \Gamma \subset G}$ ${\displaystyle \mu }$ ${\displaystyle \mu _{\Gamma }}$ ${\displaystyle \Gamma \backslash G}$ ${\displaystyle A\subset G}$ ${\displaystyle A\cap \gamma A=\emptyset \forall \gamma \in \Gamma -\left\{e\right\}}$ ${\displaystyle \mu _{\Gamma }(\pi (A))=\mu (A)}$

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). ${\displaystyle \Gamma \subset G}$ ${\displaystyle \Gamma \backslash G}$

The grid is called uniform or co-compact if it is compact. ${\displaystyle \Gamma \backslash G}$

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. ${\displaystyle \Gamma \subset G}$ ${\displaystyle G}$ ${\displaystyle G=G_{1}\times G_{2}}$ ${\displaystyle \Gamma _{1}\subset G_{1},\Gamma _{2}\subset G_{2}}$ ${\displaystyle \Gamma _{1}\times \Gamma _{2}}$ ${\displaystyle \Gamma }$

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