Lattice in a Lie group

A lattice in a Lie group $G$ is a discrete subgroup $\Gamma$ such that $\Gamma\backslash G$ admits a nonzero finite $G$ -invariant Radon measure for the right action. This is the finite-covolume condition; the invariant measure is locally finite and compatible with Haar measure.

Geometry

For $G=\operatorname{PSL}_2(\mathbb C)$ , finite covolume is equivalent to finite hyperbolic volume of $\Gamma\backslash\mathbb H^3$ . The stabilizer of an interior point is compact, so passing from $G$ to hyperbolic space preserves finiteness of quotient volume.

Distinguish the meanings of lattice

The subgroup $\mathbb Z^n\subseteq\mathbb R^n$ is a familiar example. This definition also covers nonabelian groups; it has no connection with the meet-and-join axioms of an order-theoretic lattice.

References

T. Church, B. Farb, and A. Putman, Integrality in the Steinberg module and the top-dimensional cohomology of SL_n O_K, Example 5.6, p. 31 of the linked version. Author-hosted paper Example 5.6 for the Bianchi application.