Lattice (discrete subgroup)
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 has finite invariant measure. In the special case of subgroups of Rn, this amounts to the usual 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 trees (the latter are known as tree lattices).
Lattices are of interest in many areas of mathematics: geometric group theory (as particularly nice examples of discrete groups), in differential geometry (through the construction of locally homogeneous manifolds), in number theory (through arithmetic groups), in ergodic theory (through the study of homogeneous flows on the quotient spaces) and in combinatorics (through the construction of expanding Cayley graphs and other combinatorial objects).
Any finite-index subgroup of a lattice is also a lattice in the same group. More generally, a subgroup commensurable to a lattice is a lattice.
Not every locally compact group contains a lattice, and there is no general group-theoretical sufficient condition for this. On the other hand, there are plenty of more specific settings where such criteria exist. For example, the existence or non-existence of lattices in Lie groups is a well-understood topic.
As we mentioned, a necessary condition for a group to contain a lattice is that the group must be unimodular. This allows for the easy construction of groups without lattices, for example the group of invertible upper triangular matrices or the affine groups. It is also not very hard to find unimodular groups without lattices, for example certain nilpotent Lie groups as explained below.
A stronger condition than unimodularity is simplicity. This is sufficient to imply the existence of a lattice in a Lie group, but in the more general setting of locally compact groups there exists simple groups without lattices, for example the "Neretin groups".
A nilpotent Lie group G contains a lattice if and only if the Lie algebra 𝓰 of G can be defined over the rationals. That is, if and only if the structure constants of 𝓰 are rational numbers. More precisely: In a nilpotent group whose Lie algebra has only rational structure constants, lattices are the images via the exponential map of lattices (in the more elementary sense of Lattice (group)) in the Lie algebra.
Finally, a nilpotent group is isomorphic to a lattice in a nilpotent Lie group if and only if it contains a subgroup of finite index which is torsion-free and finitely generated.
The criterion for nilpotent Lie groups to have a lattice given above does not apply to more general solvable Lie groups. It remains true that any lattice in a solvable Lie group is uniform and that lattices in solvable groups are finitely presented.
The real rank of a Lie group is the maximal dimension of an abelian subgroup containing only semisimple elements. The semisimple Lie groups of real rank 1 without compact factors are (up to isogeny) those in the following list (see List of simple Lie groups):
The real rank of a Lie group has a significant influence on the behaviour of the lattices it contains. In particular the behaviour of lattices in the first two families of groups (and to a lesser extent that of lattices in the latter two) differs much from that of irreducible lattices in groups of higher rank. For example:
The property known as (T) was introduced by Kazhdan to study the algebraic structure lattices in certain Lie groups when the classical, more geometric methods failed or at least were not as efficient. The fundamental result when studying lattices is the following:A lattice in a locally compact group has property (T) if and only if the group itself has property (T).
Using harmonic analysis it is possible to classify semisimple Lie groups according to whether or not they have the property. As a consequence we get the following result, further illustrating the dichotomy of the previous section:
Lattices in semisimple Lie groups are always finitely presented, and actually satisfy stronger finiteness conditions. For uniform lattices this is a direct consequence of cocompactness. In the non-uniform case this can be proved using reduction theory. However, for mere finite presentability a much faster proof is by using Kazhdan's property (T) when possible.
In the latter case all lattices are in fact free groups (up to finite index).
Another group of phenomena concerning lattices in semisimple algebraic groups is collectively known as rigidity. Here are three classical examples of results in this category.
Local rigidity results state that in most situations every subgroup which is sufficiently "close" to a lattice (in the intuitive sense, formalised by Chabauty topology or by the topology on a character variety) is actually conjugated to the original lattice by an element of the ambient Lie group. A consequence of local rigidity and the Kazhdan-Margulis theorem is Wang's theorem: in a given group (with a fixed Haar measure), for any v>0 there are only finitely many (up to conjugation) lattices with covolume bounded by v.
Superrigidity provides (for Lie groups and algebraic groups over local fields of higher rank) a strengthening of both local and strong rigidity, dealing with arbitrary homomorphisms from a lattice in an algebraic group G into another algebraic group H. It was proven by Grigori Margulis and is an essential ingredient in the proof of his arithmeticity theorem.
As lattices in rank-one p-adic groups are virtually free groups they are very non-rigid.