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 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 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).
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
The definition of a lattice used in mathematics relies upon the second meaning (in particular to include such examples as
Uniform lattices are quasi-isometric to their ambient groups, but non-uniform ones are not even coarsely equivalent to it.
Not every locally compact group contains a lattice, and there is no general group-theoretical sufficient condition for this.
For example, the existence or non-existence of lattices in Lie groups is a well-understood topic.
[4] 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.
Generalising the construction above one gets the notion of an arithmetic lattice in a semisimple Lie group.
a consequence of the arithmetic construction is that any semisimple Lie group contains a lattice.
(an abelian subgroup containing only semisimple elements with at least one real eigenvalue distinct from
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:[15] 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.
Interesting examples in this class of Riemannian spaces include compact flat manifolds and nilmanifolds.
: such Riemannian manifolds are called symmetric spaces of non-compact type without Euclidean factors.
This correspondence can be extended to all lattices by adding orbifolds on the geometric side.
-rank r. Then: In the latter case all lattices are in fact free groups (up to finite index).
An example of such a lattice is given by This arithmetic construction can be generalised to obtain the notion of an S-arithmetic group.
This fact makes the adèle groups very effective as tools in the theory of automorphic forms.
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
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).
In this case there are in fact continuously many lattices and they give rise to Teichmüller spaces.
In fact they are accumulation points (in the Chabauty topology) of lattices of smaller covolume, as demonstrated by hyperbolic Dehn surgery.
The discreteness in this case is easy to see from the group action on the tree: a subgroup of
Thus the more interesting tree lattices are the non-uniform ones, equivalently those for which the quotient graph
be of "finite volume" in a suitable sense (which can be expressed combinatorially in terms of the action of
), more general than the stronger condition that the quotient be finite (as proven by the very existence of nonuniform tree lattices).
