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