Lattice (discrete subgroup)

A portion of the discrete Heisenberg group, a discrete subgroup of the continuous Heisenberg Lie group. (The coloring and edges are only for visual aid.)

In Lie theory and related areas of mathematics, a lattice in a locally compact topological group is a discrete subgroup with the property that the quotient space has finite invariant measure. In the special case of subgroups of Rn, this amounts to the usual geometric notion of a lattice, and both the algebraic structure of lattices and the geometry of the totality of all lattices are relatively well understood. Deep results of Borel, Harish-Chandra, Mostow, Tamagawa, M. S. Raghunathan, Margulis, Zimmer obtained from the 1950s through the 1970s provided examples and generalized much of the theory to the setting of nilpotent Lie groups and semisimple algebraic groups over a local field. In the 1990s, Bass and Lubotzky initiated the study of tree lattices, which remains an active research area.

Definition

Let G be a locally compact topological group with the Haar measure μ. A discrete subgroup Γ is called a lattice in G if the quotient space G/Γ has finite invariant measure, that is, if G is a unimodular group and the volume μ(G/Γ) is finite. The lattice is uniform (or cocompact) if the quotient space is compact, and nonuniform otherwise.

Arithmetic lattices

An archetypical example of a nonuniform lattice is given by the group SL(2,Z), which is a lattice in the special linear group SL(2,R), and by the closely related modular group. This construction admits a far-reaching generalization to a class of lattices in all semisimple algebraic groups over a local field F called arithmetic lattices. For example, let F = R be the field of real numbers. Roughly speaking, the Lie group G(R) is formed by all matrices with entries in R satisfying certain algebraic conditions, and by restricting the entries to the integers Z, one obtains a lattice G(Z). Conversely, Grigory Margulis proved that under certain assumptions on G, any lattice in it essentially arises in this way. This remarkable statement is known as Arithmeticity of lattices or Margulis Arithmeticity Theorem.

S-arithmetic lattices

Arithmetic lattices admit an important generalization, known as the S-arithmetic lattices. The first example is given by the diagonally embedded subgroup

SL\left(2,\mathbb{Z}\left[\frac{1}{p}\right]\right) \subset SL(2,\mathbb{R})\times SL(2,\mathbb{Q}_p), S=\{p, \infty\}.

This is a lattice in the product of algebraic groups over different local fields, both real and p-adic. It is formed by the unimodular matrices of order 2 with entries in the localization of the ring of integers at the prime p. The set S is a finite set of places of Q which includes all archimedean places and the locally compact group is the direct product of the groups of points of a fixed linear algebraic group G defined over Q (or a more general global field) over the completions of Q at the places from S. To form the discrete subgroup, instead of matrices with integer entries, one considers matrices with entries in the localization over the primes (nonarchimedean places) in S. Under fairly general assumptions, this construction indeed produces a lattice. The class of S-arithmetic lattices is much wider than the class of arithmetic lattices, but they share many common features.

Adelic case

A lattice of fundamental importance for the theory of automorphic forms is given by the group G(K) of K-points of a semisimple (or reductive) linear algebraic group G defined over a global field K. This group diagonally embeds into the adelic algebraic group G(A), where A is the ring of adeles of K, and is a lattice there. Unlike arithmetic lattices, G(K) is not finitely generated.

Rigidity

Another group of phenomena concerning lattices in semisimple algebraic groups is collectively known as rigidity. The Mostow rigidity theorem showed that the algebraic structure of a lattice in simple Lie group G of split rank at least two determines G. Thus any isomorphism of lattices in two such groups is essentially induced by an isomorphism between the groups themselves. Superrigidity provides a generalization dealing with homomorphisms from a lattice in an algebraic group G into another algebraic group H.

Tree lattices

Let X be a locally finite tree. Then the automorphism group G of X is a locally compact topological group, in which the basis of the topology is given by the stabilizers of finite sets of vertices. Vertex stabilizers Gx are thus compact open subgroups, and a subgroup Γ of G is discrete if Γx is finite for some (and hence, for any) vertex x. The subgroup Γ is an X-lattice if the suitably defined volume of X/\Gamma is finite, and a uniform X-lattice if this quotient is a finite graph. In case G\backslash X is finite, this is equivalent to Γ being a lattice (respectively, a uniform lattice) in G.

See also

References