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.
Lattices are of interest in many areas of mathematics: geometric group theory, in differential geometry, in number theory, in ergodic theory and in combinatorics.

Generalities on lattices

Informal discussion

Lattices are best thought of as discrete approximations of continuous 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 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 or measure-theoretical. Note that since the Haar measure is a Borel measure, in particular 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 but the first also has its own interest.

Definition

Let be a locally compact group and a discrete subgroup. Then is called a lattice in if in addition there exists a Borel measure on the quotient space which is finite and -invariant.
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 of finite volume for the Haar measure.
A lattice is called uniform when the quotient space is compact. 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 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.
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 exists 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 then a discrete subgroup is a lattice if and only if it is not contained in a proper connected subgroup.
A nilpotent Lie group contains a lattice if and only if it can be defined over the rationals, that is if and only if its structure constants are rational numbers. More precisely, in a nilpotent group satisfying this condition lattices correspond via the exponential map to lattices in the Lie algebra.
A lattice in a nilpotent Lie group is always finitely generated ; 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 of 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 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:
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 a_i - b_i \sqrt 2.

Rank 1 versus higher rank

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 those in the following list :
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 differs much from that of irreducible lattices in groups of higher rank. For example:
The property known as 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:
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. For uniform lattices this is a direct consequence of cocompactness. In the non-uniform case this can be proved using reduction theory. However a much faster proof is by using Kazhdan's property when possible.

Riemannian manifolds associated to lattices in Lie groups

Left-invariant metrics

If is a Lie group then from an inner product on the tangent space one can construct a Riemannian metric on as follows: if belong to the tangent space at a point put where indicates the tangent map of the diffeomorphism of.
The maps for are by definition isometries for this metric. In particular, if is any discrete subgroup in the quotient is a Riemannian manifold locally isometric to with the metric.
The Riemannian volume form associated to and we see that the quotient manifold is of finite Riemannian volume if and only if is a lattice.
Interesting examples in this class of Riemannian spaces include compact flat manifolds and nilmanifolds.

Locally symmetric spaces

A natural inner product on is given by the Killing form. If is not compact it is not definite and hence not an inner product: however when is semisimple and is a maximal compact subgroup it can be used to define a -invariant metric on the homogeneous space : such Riemannian manifolds are called symmetric spaces of non-compact type without Euclidean factors.
A subgroup acts freely, properly discontinuously on if and only if it is discrete and torsion-free. The quotients are called locally symmetric spaces. There is thus a bijective correspondence between complete locally symmetric spaces locally isomorphic to and of finite Riemannian volume, and torsion-free lattices in. This correspondence can be extended to all lattices by adding orbifolds on the geometric side.

Lattices in p-adic Lie groups

A class of groups with similar properties to real semisimple Lie groups are semisimple algebraic groups over local fields of characteristic 0, for example the p-adic fields. There is an arithmetic construction similar to the real case, and the dichotomy between higher rank and rank one also holds in this case, in a more marked form. Let be an algebraic group over of split--rank r. Then:
In the latter case all lattices are in fact free groups.

S-arithmetic groups

More generally one can look at lattices in groups of the form
where is a semisimple algebraic group over. Usually is allowed, in which case is a real Lie group. An example of such a lattice is given by
This arithmetic construction can be generalised to obtain the notion of an S-arithmetic group. The Margulis arithmeticity theorem applies to this setting as well. In particular, if at least two of the factors are noncompact then any irreducible lattice in is S-arithmetic.

Lattices in adelic groups

If is a semisimple algebraic group over a number field and its adèle ring then the group of adélic points is well-defined and it is a locally compact group which naturally contains the group of -rational point as a discrete subgroup. The Borel-Harish-Chandra theorem extends to this setting, and is a lattice.
The strong approximation theorem relates the quotient to more classical S-arithmetic quotients. This fact makes the adèle groups very effective as tools in the theory of automorphic forms. In particular modern forms of the trace formula are usually stated and proven for adélic groups rather than for Lie groups.

Rigidity

Rigidity results

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 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, for any v>0 there are only finitely many lattices with covolume bounded by v.
The Mostow rigidity theorem states that for lattices in simple Lie groups not locally isomorphic to any isomorphism of lattices is essentially induced by an isomorphism between the groups themselves. In particular, a lattice in a Lie group "remembers" the ambient Lie group through its group structure. The first statement is sometimes called strong rigidity and is due to George Mostow and Gopal Prasad.
Superrigidity provides a generalization dealing with 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.

Nonrigidity in low dimensions

The only groups for which Mostow rigidity does not hold are all groups locally isomorphic to. In this case there are in fact continuously many lattices and they give rise to Teichmüller spaces.
Nonuniform lattices in the group are not locally rigid. In fact they are accumulation points of lattices of smaller covolume, as demonstrated by hyperbolic Dehn surgery.
As lattices in rank-one p-adic groups are virtually free groups they are very non-rigid.

Tree lattices

Definition

Let be a tree with a cocompact group of automorphisms; for example, can be a regular or biregular tree. The group of automorphisms of is a locally compact group. Any group which is a lattice in some is then called a tree lattice.
The discreteness in this case is easy to see from the group action on the tree: a subgroup of is discrete if and only if all vertex stabilisers are finite groups.
It is easily seen from the basic theory of group actions on trees that uniform tree lattices are virtually free groups. Thus the more interesting tree lattices are the non-uniform ones, equivalently those for which the quotient graph is infinite. The existence of such lattices is not easy to see.

Tree lattices from algebraic groups

If is a local field of positive characteristic and an algebraic group defined over of -split rank one, then any lattice in is a tree lattice through its action on the Bruhat–Tits building which in this case is a tree. In contrast to the characteristic 0 case such lattices can be nonuniform, and in this case they are never finitely generated.

Tree lattices from Bass–Serre theory

If is the fundamental group of an infinite graph of groups, all of whose vertex groups are finite, and under additional necessary assumptions on the index of the edge groups and the size of the vertex groups, then the action of on the Bass-Serre tree associated to the graph of groups realises it as a tree lattice.

Existence criterion

More generally one can ask the following question: if is a closed subgroup of, under which conditions does contain a lattice? The existence of a uniform lattice is equivalent to being unimodular and the quotient being finite. The general existence theorem is more subtle: it is necessary and sufficient that be unimodular, and that the quotient be of "finite volume" in a suitable sense, more general than the stronger condition that the quotient be finite.