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Definition
A lattice with compact quotient; a nonuniform lattice has noncompact quotient.
A lattice is uniform, or cocompact, if is compact. It is nonuniform if this quotient is noncompact.
Both uniform and nonuniform lattices have finite covolume. A discrete subgroup with infinite covolume is not a nonuniform lattice: it is not a lattice at all.
For , compactness is equivalent to compactness of . Bianchi groups give nonuniform lattices.
A lattice in a Lie group is a discrete subgroup such that admits a nonzero finite -invariant Radon measure for the right action. This is the finite-covolume condition; the invariant measure is locally finite and compatible with Haar measure.
A compact set is a subset of a topological space such that every open cover of contains a finite subcover: whenever is a family of open sets with , there exist indices such that
Every Bianchi group is a nonuniform lattice in . Thus it is discrete, and its hyperbolic quotient is noncompact with finite volume.