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 R"n", 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 formula_1 of integer vectors "looks like" the real vector space formula_2 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 formula_3 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 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 (in particular to include such examples as formula_4) but the first also has its own interest (such lattices are called uniform).
Definition.
Let formula_5 be a locally compact group and formula_6 a discrete subgroup (this means that there exists a neighbourhood formula_7 of the identity element formula_8 of formula_5 such that formula_10). Then formula_6 is called a lattice in formula_5 if in addition there exists a Borel measure formula_13 on the quotient space formula_14 which is finite (i.e. formula_15) and formula_5-invariant (meaning that for any formula_17 and any open subset formula_18 the equality formula_19 is satisfied).
A slightly more sophisticated formulation is as follows: suppose in addition that formula_5 is unimodular, then since formula_6 is discrete it is also unimodular and by general theorems there exists a unique formula_5-invariant Borel measure on formula_14 up to scaling. Then formula_6 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 formula_5 being a lattice is equivalent to it having a fundamental domain (for the action on formula_5 by left-translations) of finite volume for the Haar measure.
A lattice formula_27 is called uniform when the quotient space formula_28 is compact (and "non-uniform" otherwise). Equivalently a discrete subgroup formula_27 is a uniform lattice if and only if there exists a compact subset formula_30 with formula_31. Note that if formula_6 is any discrete subgroup in formula_5 such that formula_34 is compact then formula_6 is automatically a lattice in formula_5.
First examples.
The fundamental, and simplest, example is the subgroup formula_1 which is a lattice in the Lie group formula_2. A slightly more complicated example is given by the discrete Heisenberg group inside the continuous Heisenberg group.
If formula_5 is a discrete group then a lattice in formula_5 is exactly a subgroup formula_6 of finite index (i.e. the quotient set formula_28 is finite).
All of these examples are uniform. A non-uniform example is given by the modular group formula_43 inside formula_44, and also by the higher-dimensional analogues formula_45.
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 formula_46 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 "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.
A lattice in a nilpotent Lie group formula_46 is always finitely generated (and hence finitely presented since it is itself nilpotent); in fact it is generated by at most formula_48 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 formula_5 is a semisimple linear algebraic group in formula_50 which is defined over the field formula_51 of rational numbers (i.e. the polynomial equations defining formula_5 have their coefficients in formula_51) then it has a subgroup formula_54. A fundamental theorem of Armand Borel and Harish-Chandra states that formula_6 is always a lattice in formula_5; the simplest example of this is the subgroup formula_4.
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 formula_51 a consequence of the arithmetic construction is that any semisimple Lie group contains a lattice.
Irreducibility.
When the Lie group formula_5 splits as a product formula_60 there is an obvious construction of lattices in formula_5 from the smaller groups: if formula_62 are lattices then formula_63 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 formula_64 is the decomposition of formula_5 into simple factors, a lattice formula_27 is said to be irreducible if either of the following equivalent conditions hold: 
An example of an irreducible lattice is given by the subgroup formula_71 which we view as a subgroup formula_72 via the map formula_73 where formula_74 is the Galois map sending a matric with coefficients formula_75 to formula_76.
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 (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:
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:
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:
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. However, for mere finite presentability a much faster proof is by using Kazhdan's property (T) when possible.
Riemannian manifolds associated to lattices in Lie groups.
Left-invariant metrics.
If formula_5 is a Lie group then from an inner product formula_94 on the tangent space formula_95 (the Lie algebra of formula_5) one can construct a Riemannian metric on formula_5 as follows: if formula_98 belong to the tangent space at a point formula_99 put formula_100 where formula_101 indicates the tangent map (at formula_102) of the diffeomorphism formula_103 of formula_5.
The maps formula_105 for formula_99 are by definition isometries for this metric formula_107. In particular, if formula_6 is any discrete subgroup in formula_5 (so that it acts freely and properly discontinuously by left-translations on formula_5) the quotient formula_111 is a Riemannian manifold locally isometric to formula_5 with the metric formula_107.
The Riemannian volume form associated to formula_114 defines a Haar measure on formula_5 and we see that the quotient manifold is of finite Riemannian volume if and only if formula_6 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 formula_95 is given by the Killing form. If formula_5 is not compact it is not definite and hence not an inner product: however when formula_5 is semisimple and formula_120 is a maximal compact subgroup it can be used to define a formula_5-invariant metric on the homogeneous space formula_122: such Riemannian manifolds are called symmetric spaces of non-compact type without Euclidean factors.
A subgroup formula_27 acts freely, properly discontinuously on formula_124 if and only if it is discrete and torsion-free. The quotients formula_125 are called locally symmetric spaces. There is thus a bijective correspondence between complete locally symmetric spaces locally isomorphic to formula_124 and of finite Riemannian volume, and torsion-free lattices in formula_5. 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 (with respect to lattices) to real semisimple Lie groups are semisimple algebraic groups over local fields of characteristic 0, for example the p-adic fields formula_128. 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 formula_5 be an algebraic group over formula_128 of split-formula_128-rank "r". Then:
In the latter case all lattices are in fact free groups (up to finite index).
S-arithmetic groups.
More generally one can look at lattices in groups of the form
where formula_134 is a semisimple algebraic group over formula_128. Usually formula_136 is allowed, in which case formula_137 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 formula_134 are noncompact then any irreducible lattice in formula_5 is S-arithmetic.
Lattices in adelic groups.
If formula_141 is a semisimple algebraic group over a number field formula_142 and formula_143 its adèle ring then the group formula_144 of adélic points is well-defined (modulo some technicalities) and it is a locally compact group which naturally contains the group formula_145 of formula_142-rational point as a discrete subgroup. The Borel–Harish-Chandra theorem extends to this setting, and formula_147 is a lattice.
The strong approximation theorem relates the quotient formula_148 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 (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".
The Mostow rigidity theorem states that for lattices in simple Lie groups not locally isomorphic to formula_44 (the group of 2 by 2 matrices with determinant 1) 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 (Mostow proved it for cocompact lattices and Prasad extended it to the general case).
"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.
Nonrigidity in low dimensions.
The only semisimple Lie groups for which Mostow rigidity does not hold are all groups locally isomorphic to formula_150. In this case there are in fact continuously many lattices and they give rise to Teichmüller spaces.
Nonuniform lattices in the group formula_151 are not locally rigid. In fact they are accumulation points (in the Chabauty topology) 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 formula_152 be a tree with a cocompact group of automorphisms; for example, formula_152 can be a regular or biregular tree. The group of automorphismsformula_154 of formula_152 is a locally compact group (when endowed with the compact-open topology, in which a basis of neighbourhoods of the identity is given by the stabilisers of finite subtrees, which are compact). Any group which is a lattice in some formula_154 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 formula_154 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 formula_158 is infinite. The existence of such lattices is not easy to see.
Tree lattices from algebraic groups.
If formula_142 is a local field of positive characteristic (i.e. a completion of a function field of a curve over a finite field, for example the field of formal Laurent power series formula_160) and formula_5 an algebraic group defined over formula_142 of formula_142-split rank one, then any lattice in formula_5 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 formula_6 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 formula_6 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 formula_167 is a closed subgroup of formula_154, under which conditions does formula_167 contain a lattice? The existence of a uniform lattice is equivalent to formula_167 being unimodular and the quotient formula_171 being finite. The general existence theorem is more subtle: it is necessary and sufficient that formula_167 be unimodular, and that the quotient formula_171 be of "finite volume" in a suitable sense (which can be expressed combinatorially in terms of the action of formula_167), more general than the stronger condition that the quotient be finite (as proven by the very existence of nonuniform tree lattices).