covolume

Covolume is a term used in geometry and group theory to describe the “size” of a lattice inside a space, measured as the volume of the fundamental repeating unit or, more generally, of a quotient space. It appears in two related contexts: lattices in Euclidean space and lattices inside Lie groups.

In Euclidean space, let Λ be a lattice in R^n (a discrete subgroup generated by n linearly independent vectors). The covolume of Λ is the volume of a fundamental domain for the action of Λ on R^n, equivalently the volume of the parallelepiped spanned by a chosen basis of Λ. This value is independent of the chosen basis and equals the absolute value of the determinant of the basis matrix. Practically, it measures the density of lattice points: smaller covolume means a denser lattice. For a lattice generated by scaling, the covolume scales by the corresponding power of the scaling factor (e.g., covolume of λΛ is |λ|^n times the covolume of Λ).

In Lie groups, let G be a Lie group with a Haar measure μ. A lattice Γ ⊂ G is a discrete subgroup such that the quotient G/Γ has finite μ-measure. The finite value μ(G/Γ) is called the covolume of Γ in G; it depends on the chosen Haar measure. This general notion includes, for example, lattices in SL(2,R) and the resulting finite-volume quotients of hyperbolic space.

Covolume is closely tied to the concept of a fundamental domain and to questions about density, volume, and symmetry in geometry and number theory. Examples include the unit cell volume of a crystal lattice and the finite volume of modular surfaces arising from arithmetic lattices.