Lattice (discrete subgroup)

A portion of the discrete Heisenberg group, a discrete subgroup of the continuous Heisenberg Lie group. (The coloring and edges are only for visual aid.)

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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 ${\displaystyle \mathbb {Z} ^{n}}$ of integer vectors "looks like" the real vector space ${\displaystyle \mathbb {R} ^{n}}$ 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 ${\displaystyle \mathbb {Z} ^{n}\subset \mathbb {R} ^{n}}$ 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 ${\displaystyle \mathrm {SL} _{2}(\mathbb {Z} )\subset \mathrm {SL} _{2}(\mathbb {R} )}$) 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 ${\displaystyle G}$ be a locally compact group and ${\displaystyle \Gamma }$ a discrete subgroup (this means that there exists a neighbourhood ${\displaystyle U}$ of the identity element ${\displaystyle e_{G}}$ of ${\displaystyle G}$ such that ${\displaystyle \Gamma \cap U=\{e_{G}\}}$). Then ${\displaystyle \Gamma }$ is called a lattice in ${\displaystyle G}$ if in addition there exists a Borel measure ${\displaystyle \mu }$ on the quotient space ${\displaystyle G/\Gamma }$ which is finite (i.e. ${\displaystyle \mu (G/\Gamma )<+\infty }$) and ${\displaystyle G}$-invariant (meaning that for any ${\displaystyle g\in G}$ and any open subset ${\displaystyle W\subset G/\Gamma }$ the equality ${\displaystyle \mu (gW)=\mu (W)}$ is satisfied).

A slightly more sophisticated formulation is as follows: suppose in addition that ${\displaystyle G}$ is unimodular, then since ${\displaystyle \Gamma }$ is discrete it is also unimodular and by general theorems there exists a unique ${\displaystyle G}$-invariant Borel measure on ${\displaystyle G/\Gamma }$ up to scaling. Then ${\displaystyle \Gamma }$ 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 ${\displaystyle G}$ being a lattice is equivalent to it having a fundamental domain (for the action on ${\displaystyle G}$ by left-translations) of finite volume for the Haar measure.

A lattice ${\displaystyle \Gamma \subset G}$ is called uniform when the quotient space ${\displaystyle G/\Gamma }$ is compact (and non-uniform otherwise). Equivalently a discrete subgroup ${\displaystyle \Gamma \subset G}$ is a uniform lattice if and only if there exists a compact subset ${\displaystyle C\subset G}$ with ${\displaystyle G=\bigcup {}_{\gamma \in \Gamma }\,C\gamma }$. Note that if ${\displaystyle \Gamma }$ is any discrete subgroup in ${\displaystyle G}$ such that ${\displaystyle G/\Gamma }$ is compact then ${\displaystyle \Gamma }$ is automatically a lattice in ${\displaystyle G}$.

First examples[]

The fundamental, and simplest, example is the subgroup ${\displaystyle \mathbb {Z} ^{n}}$ which is a lattice in the Lie group ${\displaystyle \mathbb {R} ^{n}}$. A slightly more complicated example is given by the discrete Heisenberg group inside the continuous Heisenberg group.

If ${\displaystyle G}$ is a discrete group then a lattice in ${\displaystyle G}$ is exactly a subgroup ${\displaystyle \Gamma }$ of finite index (i.e. the quotient set ${\displaystyle G/\Gamma }$ is finite).

All of these examples are uniform. A non-uniform example is given by the modular group ${\displaystyle \mathrm {SL} _{2}(\mathbb {Z} )}$ inside ${\displaystyle \mathrm {SL} _{2}(\mathbb {R} )}$, and also by the higher-dimensional analogues ${\displaystyle \mathrm {SL} _{n}(\mathbb {Z} )\subset \mathrm {SL} _{n}(\mathbb {R} )}$.

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".^[1]

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 ${\displaystyle N}$ 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^[2] (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