Going in the other direction, if G is any topological group and K is a discrete normal subgroup of G then the quotient map p : G → G / K is a covering homomorphism.
If G is connected then K, being a discrete normal subgroup, necessarily lies in the center of G and is therefore abelian.
Let H be a topological group and let G be a covering space of H. If G and H are both path-connected and locally path-connected, then for any choice of element e* in the fiber over e ∈ H, there exists a unique topological group structure on G, with e* as the identity, for which the covering map p : G → H is a homomorphism.
By the path-lifting property of covering spaces there is a unique lift of h to G with initial point e*.
One must show that this definition is independent of the choice of paths f and g, and also that the group operations are continuous.
Alternatively, the group law on G can be constructed by lifting the group law H × H → H to G, using the lifting property of the covering map G × G → H × H. The non-connected case is interesting and is studied in the papers by Taylor and by Brown-Mucuk cited below.
The universal cover of H is given as the quotient of PH by the normal subgroup of null-homotopic loops.
The projection PH → H descends to the quotient giving the covering map.
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.
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
The preimage of SL2(Z) in the universal cover is isomorphic to the braid group on three strands.