Arithmetic Lattice
In Lie theory and related areas of mathematics, a lattice in a locally compact group is a discrete subgroup with the property that the Quotient space (topology), quotient space has finite invariant measure. In the special case of subgroups of R''n'', this amounts to the usual Lattice (group), geometric notion of a lattice as a periodic subset of points, and both the algebraic structure of lattices and the geometry of the space of all lattices are relatively well understood. The theory is particularly rich for lattices in semisimple Lie groups or more generally in semisimple algebraic groups over local fields. In particular there is a wealth of rigidity results in this setting, and a celebrated theorem of Grigory Margulis states that in most cases all lattices are obtained as arithmetic groups. Lattices are also well-studied in some other classes of groups, in particular groups associated to Kac–Moody algebras and automorphisms groups of regular Tree (graph theory), trees (the ...
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