Arithmeticity

Arithmeticity is a term used in the study of lattices in Lie groups and algebraic groups. It describes when a discrete subgroup with finite covolume, called a lattice, is defined by arithmetic data from number theory.

Definition. Let G be a linear algebraic group defined over a number field k. For a finite set of places S of k, let O_S be the ring of S-integers. The group G(O_S) is an arithmetic subgroup of the real Lie group G(R) (via the embeddings of k into R or C). A discrete subgroup Γ of G(R) is called arithmetic if it is commensurable with G(O_S) for some choice of G, k, and S. When Γ is a lattice, it is referred to as an arithmetic lattice.

Examples. The classical example is SL_n(Z) as a lattice in SL_n(R). More generally, for any number field k and algebraic group G defined over k, the group G(O_k) (and its S-arithmetic variants G(O_S)) yields arithmetic lattices inside the corresponding real Lie groups G(R).

Theorems and contrasts. A central result is Margulis’s arithmeticity theorem: if G is a connected semisimple algebraic group defined over a number field and Γ is an irreducible lattice in G(R) with real rank at least 2, then Γ is arithmetic. In rank-one groups (for example certain lattices in SO(n,1) and SU(n,1)), non-arithmetic lattices exist, so arithmeticity is not automatic there. The notion extends to S-arithmetic groups, which generalize arithmetic lattices by allowing several completions of k.

Significance. Arithmeticity connects discrete subgroups to number theory and algebraic groups, underpinning rigidity phenomena and influencing the structure and classification of lattices in higher-rank Lie groups.