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).
Generalities on lattices
Informal discussion
Lattices are best thought of as discrete approximations of continuous groups (such as Lie groups). For example, it is intuitively clear that the subgroup of integer vectors "looks like" the real vector space in some sense, while both groups are essentially different: one is finitely generated and countable, while the other is not finitely generated and has the cardinality of the continuum.
Rigorously defining the meaning of "approximation of a continuous group by a discrete subgroup" in the previous paragraph in order to get a notion generalising the example is a matter of what it is designed to achieve. Maybe the most obvious idea is to say that a subgroup "approximates" a larger group is that the larger group can be covered by the translates of a "small" subset by all elements in the subgroups. In a locally compact topological group there are two immediately available notions of "small": topological (a compact, or relatively compact subset) or measure-theoretical (a subset of finite Haar measure). Note that since the Haar measure is a Radon measure, so it gives finite mass to compact subsets, the second definition is more general. The definition of a lattice used in mathematics relies upon the second meaning (in particular to include such examples as ) but the first also has its own interest (such lattices are called uniform).
Other notions are coarse equivalence and the stronger quasi-isometry. Uniform lattices are quasi-isometric to their ambient groups, but non-uniform ones are not even coarsely equivalent to it.
Definition
Let be a locally compact group and a discrete subgroup (this means that there exists a neighbourhood of the identity element of such that ). Then is called a lattice in if in addition there exists a Borel measure on the quotient space which is finite (i.e. ) and -invariant (meaning that for any and any open subset the equality is satisfied).
A slightly more sophisticated formulation is as follows: suppose in addition that is unimodular, then since is discrete it is also unimodular and by general theorems there exists a unique -invariant Borel measure on up to scaling. Then is a lattice if and only if this measure is finite.
In the case of discrete subgroups this invariant measure coincides locally with the Haar measure and hence a discrete subgroup in a locally compact group being a lattice is equivalent to it having a fundamental domain (for the action on by left-translations) of finite volume for the Haar measure.
A lattice is called uniform (or cocompact) when the quotient space is compact (and non-uniform otherwise). Equivalently a discrete subgroup is a uniform lattice if and only if there exists a compact subset with . Note that if is any discrete subgroup in such that is compact then is automatically a lattice in .
First examples
The fundamental, and simplest, example is the subgroup which is a lattice in the Lie group . A slightly more complicated example is given by the discrete Heisenberg group inside the continuous Heisenberg group.
If is a discrete group then a lattice in is exactly a subgroup of finite index (i.e. the quotient set is finite).
All of these examples are uniform. A non-uniform example is given by the modular group inside , and also by the higher-dimensional analogues .
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.
Which groups have lattices?
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 exist simple groups without lattices, for example the "Neretin groups".
Lattices in solvable Lie groups
Nilpotent Lie groups
For nilpotent groups the theory simplifies much from the general case, and stays similar to the case of Abelian groups. All lattices in a nilpotent Lie group are uniform, and if is a connected simply connected nilpotent Lie group (equivalently it does not contain a nontrivial compact subgroup) then a discrete subgroup is a lattice if and only if it is not contained in a proper connected subgroup (this generalises the fact that a discrete subgroup in a vector space is a lattice if and only if it spans the vector space).
A nilpotent Lie group contains a lattice if and only if the Lie algebra of can be defined over the rationals. That is, if and only if the structure constants of are rational numbers. More precisely: if is a nilpotent simply connected Lie group whose Lie algebra has only rational structure constants, and is a lattice in (in the more elementary sense of Lattice (group)) then generates a lattice in ; conversely, if is a lattice in then generates a lattice in .
A lattice in a nilpotent Lie group is always finitely generated (and hence finitely presented since it is itself nilpotent); in fact it is generated by at most elements.
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 general case
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.
Not all finitely generated solvable groups are lattices in a Lie group. An algebraic criterion is that the group be polycyclic.
Lattices in semisimple Lie groups
Arithmetic groups and existence of lattices
If is a semisimple linear algebraic group in which is defined over the field of rational numbers (i.e. the polynomial equations defining have their coefficients in ) then it has a subgroup . A fundamental theorem of Armand Borel and Harish-Chandra states that is always a lattice in ; the simplest example of this is the subgroup .
Generalising the construction above one gets the notion of an arithmetic lattice in a semisimple Lie group. Since all semisimple Lie groups can be defined over a consequence of the arithmetic construction is that any semisimple Lie group contains a lattice.
Irreducibility
When the Lie group splits as a product there is an obvious construction of lattices in from the smaller groups: if are lattices then is a lattice as well. Roughly, a lattice is then said to be irreducible if it does not come from this construction.
More formally, if is the decomposition of into simple factors, a lattice is said to be irreducible if either of the following equivalent conditions hold: The projection of to any factor is dense; The intersection of with any factor is not a lattice.
An example of an irreducible lattice is given by the subgroup which we view as a subgroup via the map where is the Galois map sending a matric with coefficients to .
Rank 1 versus higher rank
The real rank of a Lie group is the maximal dimension of a -split torus of (an abelian subgroup containing only semisimple elements with at least one real eigenvalue distinct from ). 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 orthogonal groups of real quadratic forms of signature for ; The unitary groups of Hermitian forms of signature for ; The groups (groups of matrices with quaternion coefficients which preserve a "quaternionic quadratic form" of signature ) for ; The exceptional Lie group (the real form of rank 1 corresponding to the exceptional Lie algebra ).
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:
There exists non-arithmetic lattices in all groups , in , and possibly in (the last is an open question) but all irreducible lattices in the others are arithmetic; Lattices in rank 1 Lie groups have infinite, infinite index normal subgroups while all normal subgroups of irreducible lattices in higher rank are either of finite index or contained in their center; Conjecturally, arithmetic lattices in higher-rank groups have the congruence subgroup property but there are many lattices in which have non-congruence finite-index subgroups.
Kazhdan's property (T)
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 do not have Kazhdan's property (T) while irreducible lattices in all other simple Lie groups do;
Finiteness properties
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. It is easier to prove finite presentability for groups with Property (T); however, there is a geometric proof which works for all semisimple Lie groups.
Riemannian manifolds associated to lattices in Lie groups
Left-invariant metrics