Covering Group - Lattice of Covering Groups

Lattice of Covering Groups

As the above suggest, if a group has a universal covering group (if it is path-connected, locally path-connected, and semilocally simply connected), with discrete center, then the set of all topological groups that are covered by the universal covering group form a lattice, corresponding to the lattice of subgroups of the center of the universal covering group: inclusion of subgroups corresponds to covering of quotient groups. The maximal element is the universal covering group while the minimal element is the universal covering group mod its center, .

This corresponds algebraically to the universal perfect central extension (called "covering group", by analogy) as the maximal element, and a group mod its center as minimal element.

This is particularly important for Lie groups, as these groups are all the (connected) realizations of a particular Lie algebra. For many Lie groups the center is the group of scalar matrices, and thus the group mod its center is the projectivization of the Lie group. These covers are important in studying projective representations of Lie groups, and spin representations lead to the discovery of spin groups: a projective representation of a Lie group need not come from a linear representation of the group, but does come from a linear representation of some covering group, in particular the universal covering group. The finite analog led to the covering group or Schur cover, as discussed above.

A key example arises from SL₂(R), which has center {±1} and fundamental group Z. It is a double cover of the centerless projective special linear group PSL₂(R), which is obtained by taking the quotient by the center. By Iwasawa decomposition, both groups are circle bundles over the complex upper half-plane, and their universal cover is a real line bundle over the half-plane that forms one of Thurston's eight geometries. Since the half-plane is contractible, all bundle structures are trivial. The preimage of SL₂(Z) in the universal cover is isomorphic to the braid group on three strands.