# Discrete subgroup

$\DeclareMathOperator{\PSL}{PSL}$

A subgroup $\Gamma$ of a topological group $G$ (in particular, a subgroup of a Lie group) which is a discrete subset of the topological space $G$. In locally compact topological groups (in particular, in Lie groups) one distinguishes lattices — i.e. discrete subgroups for which the quotient space $\Gamma\setminus G$ has finite volume in the sense of the measure induced by the left-invariant Haar measure on the group $G$. The concept of lattices includes that of uniform discrete subgroups, for which the quotient space $\Gamma\setminus G$ is compact.

If $K$ is a compact subgroup of a locally compact topological group $G$, a subgroup $\Gamma \subset G$ is discrete if and only if it is a discrete group of transformations of the space $X=G/K$ (in the sense of the action induced by the natural action of the group $G$ on $X$). Here, $\Gamma$ is a lattice (a uniform discrete subgroup) if and only if the quotient space $\Gamma \setminus X$ has finite volume (is compact) in the sense of the measure induced by the $G$-invariant measure on $X$. This makes it possible to utilize geometric methods when studying discrete subgroups of Lie groups.

One of the principal problems in the theory of discrete subgroups of Lie groups is the classification of such subgroups up to commensurability. Two subgroups $\Gamma_1$ and $\Gamma_2$ are said to be commensurable if $\Gamma_1 \cap \Gamma_2$ has finite index both in $\Gamma_1$ and in $\Gamma_2$. If one of two commensurable subgroups of a locally compact topological group is a discrete subgroup (or a lattice, or a uniform discrete subgroup), so is the other.

Up to the middle of the 20th century one basically studied individual classes of discrete subgroups of Lie groups occurring in arithmetic, function theory and physics. Historically, the first non-trivial discrete subgroup — the subgroup $\SL_2(\mathbf{Z})$ of the group $\SL_2(\mathbf{R})$, subsequently named the Kleinian modular group — was in fact studied by J.L. Lagrange and C.F. Gauss in the context of the arithmetic of quadratic forms in two variables. The subgroup $\SL_n(\mathbf{Z})$ of $\SL_n(\mathbf{R})$ is its natural generalization. The study of this group as a discrete group of transformations of the space of positive-definite quadratic forms in $n$ variables formed the subject of reduction theory, developed by A.N. Korkin, E.I. Zolotarev, Ch. Hermite, H. Minkowski, and others in the second half of the nineteenth and in the beginning of the 20th century. A series of arithmetically definable discrete subgroups of classical Lie groups — groups of units of quadratic forms with rational coefficients, groups of units of simple algebras over $\mathbf{Q}$, groups of integral symplectic matrices — were studied by C.L. Siegel in the 1940s. He proved, in particular, that all these groups are lattices in the respective Lie groups.

In the theory of functions of a complex variable the integration of algebraic functions and, more generally, the solution of differential equations with algebraic coefficients, resulted in the study of certain special functions (subsequently named automorphic functions, cf. Automorphic function) which are invariant with respect to various discrete groups consisting of transformations of the form

$$z \mapsto \frac{az+b}{cz+d}, \qquad z \in \mathbf{C}, \qquad \begin{bmatrix} a & b \\ c & d \end{bmatrix} \in \SL_2(\mathbf{R}).$$ Certain discrete subgroups of $\SL_2(\mathbf{R})$ were studied in the mid-19th century by Hermite, R. Dedekind and I.L. Fuchs. They also included the group $\SL_2(\mathbf{Z})$ (though represented differently from the presentation used by Lagrange and Gauss). A wide class of such groups, including the group $\SL_2(\mathbf{Z})$ and certain subgroups of $\SL_2(\mathbf{R})$ commensurable with it, were studied by F. Klein. Almost simultaneously (1881–1882) H. Poincaré gave a geometric description of all discrete groups consisting of transformations of the form (1). He named these groups Fuchsian groups (cf. Fuchsian group).

In the first half of the 20th century studies were made of individual classes of automorphic functions in several variables. These functions were connected with certain arithmetically definable discrete subgroups of the group $\left(\SL_2(\mathbf{R})\right)^k$ (Hilbert's modular functions), $\Sp_{2n}(\mathbf{R})$ (Siegel's modular functions) and other semi-simple Lie groups.

Since the late 19th century, crystallographic studies have centred on the symmetry groups of crystallographic lattices, which are identical with uniform discrete subgroups of the group of motions of three-dimensional Euclidean space. These, together with the related groups of motions of $n$-dimensional Euclidean space (the so-called crystallographic groups, cf. Crystallographic group) were studied in 1911 by L. Bieberbach from the algebraic point of view. He demonstrated, in particular, the theorem according to which any crystallographic group contains a uniform discrete subgroup of parallel translations.

All these studies provided the initial material for the general theory of discrete subgroups of Lie groups, the foundations of which were laid in the 1950s and 1960s.

An exhaustive theory of discrete subgroups of nilpotent Lie groups has been constructed . Its main statements are listed below:

1. If $H$ is a unipotent algebraic group defined over $\mathbf{Q}$, then the group $H_\mathbf{Z}$ of its integer points is a uniform discrete subgroup in the group $H_\mathbf{R}$ of its real points. (Here $H_\mathbf{R}$ is a simply-connected nilpotent Lie group.)
2. Any uniform discrete subgroup $\Gamma$ of a simply-connected nilpotent Lie group $G$ is arithmetic in the sense that there exist a unipotent algebraic group $H$ defined over $\mathbf{Q}$ and an isomorphism $\phi: H_\mathbf{R} \to G$ such that the subgroup $\Gamma$ is commensurable with $\phi(H_\mathbf{Z})$.
3. If $\Gamma_1$, $\Gamma_2$ are uniform discrete subgroups of simply-connected nilpotent Lie groups $G_1$ and $G_2$ respectively, then any isomorphism $\Gamma_1 \to \Gamma_2$ can be uniquely extended to an isomorphism $G_1 \to G_2$.
4. An abstract group $\Gamma$ is imbeddable as a uniform discrete subgroup in a simply-connected nilpotent Lie group if and only if $\Gamma$ is a finitely-generated torsion-free nilpotent group.

Discrete subgroups of solvable Lie groups have been fairly thoroughly studied, but the results are less complete than those obtained for nilpotent groups. Any lattice in a solvable Lie group is a uniform discrete subgroup. If $\Gamma$ is a lattice in a simply-connected solvable Lie group $G$, then $G$ has a faithful matrix representation in which the elements of $\Gamma$ are represented by integer matrices . This statement may be regarded as a generalization of Mal'tsev's theorem 2) above. The following theorem is the analogue of theorem 4). Any lattice in a simply-connected solvable Lie group is a strictly polycyclic group; conversely, any strictly polycyclic group has a subgroup of finite index which is isomorphic to a lattice in a simply-connected solvable Lie group.

The most precise results in the theory of discrete subgroups of Lie groups concern discrete subgroups of non-solvable and, in particular, semi-simple Lie groups. In  the following theorem was demonstrated, which includes, as special cases, Mal'tsev's theorem 1), the Dirichlet theorem on the units of an algebraic number field and Siegel's results (see above) on certain arithmetic discrete subgroups of semi-simple Lie groups. Let $H$ be a linear algebraic group defined over $\mathbf{Q}$. For the subgroup $H_\mathbf{Z}$ to be a lattice in $H_\mathbf{R}$ it is necessary and sufficient for $H$ not to permit rational homomorphisms into the group $\mathbf{C}^*$, defined over $\mathbf{Q}$ (this condition is satisfied, for example, if $H$ is semi-simple or unipotent). For the subgroup $H_\mathbf{Z}$ to be a uniform discrete subgroup in $H_\mathbf{R}$ it is necessary and sufficient, in addition, that all unipotent elements of the group $H_\mathbf{Q}$ lie in $U_\mathbf{Q}$, where $U$ is the unipotent radical of $H$.

The arithmeticity theorem  which follows is the analogue of theorem 2) for discrete subgroups of semi-simple Lie groups. Let $\Gamma$ be a lattice in a connected semi-simple Lie group $G$ without compact factors, and let (for the sake of convenience in formulation) the centre of $G$ be trivial. Moreover, let the lattice $\Gamma$ be irreducible in the sense that $G$ cannot be non-trivially decomposed into a direct product $G_1 \times G_2$ so that $\Gamma$ is commensurable with a subgroup of the form $\Gamma_1 \times \Gamma_2$ where $\Gamma_1 \subset G_1$ and $\Gamma_2 \subset G_2$. Then, if the real rank of $G$ exceeds one, the group $\Gamma$ is arithmetic in the sense that there exist a semi-simple algebraic group $H$, defined over $\mathbf{Q}$, and a homomorphism $\phi: H_\mathbf{R}^0 \to G$ (where $H_\mathbf{R}^0$ is the connected component of the unit of the group $H_\mathbf{R}$) such that the kernel of the homomorphism $\phi$ is compact and the subgroup $\Gamma$ is commensurable with $\phi(H_\mathbf{Z})$. The assumption that the real rank of $G$ exceeds one is essential. It is known that the theorem is invalid for the group $\PSL_2(\mathbf{R})$ (the group of motions of the Lobachevskii plane), which on the whole plays an important role in the theory of discrete subgroups of Lie groups, and also for the groups of motions of the three-, four- and five-dimensional Lobachevskii spaces , .

The strong rigidity theorem which follows is the analogue of theorem 3) for discrete subgroups of semi-simple Lie groups. Let $\Gamma_1$, $\Gamma_2$ be irreducible lattices in connected semi-simple Lie groups $G_1$, $G_2$ without compact factors, and let the centres of $G_1$, $G_2$ be trivial. Then, if $G_1$ and $G_2$ are not isomorphic to $\PSL_2(\mathbf{R})$, any isomorphism $\Gamma_1 \to \Gamma_2$ can be uniquely extended to an isomorphism $G_1 \to G_2$ , . Historically, the proof of this theorem was preceded by the proof of the weak rigidity theorem

on the extension of isomorphisms which are sufficiently close to the identity (if $G_1 = G_2$). One consequence of the weak rigidity theorem is the existence of a basis in which the elements of a discrete subgroup are written in the form of algebraic numbers. This fact played an important role in the development of the theory of discrete subgroups of semi-simple Lie groups.

Regarding discrete subgroups of the group $\PSL_2(\mathbf{R})$ see Fuchsian group.

Of the other general theorems about discrete subgroups of semi-simple Lie groups one may mention Borel's density theorem and Wang's maximality theorem. Let $\Gamma$ be a lattice in a connected semi-simple Lie group $G$ which has no compact factors. Then $\Gamma$ is dense in $G$ in the Zariski topology , and is contained in only a finite number of lattices in $G$ .

The description of lattices in arbitrary Lie groups can be reduced, to some extent, to the description of lattices in semi-simple Lie groups, in view of theorems analogous to the Bieberbach theorem on crystallographic groups mentioned above. One says that a normal subgroup $N$ of a Lie group $G$ has the Bieberbach property if for any lattice $\Gamma$ in $G$ the subgroup $N\Gamma$ is closed (and, in such a case, $N \cap \Gamma$ is automatically a lattice in $N$, while $\Gamma / N\cap \Gamma$ is a lattice in $G/N$). Bieberbach's theorem says that, in the group of motions of Euclidean space, the subgroup of parallel translations has the Bieberbach property. There exists a generalization of this theorem to Lie groups which are extensions of a simply-connected nilpotent Lie group by a compact group . Another theorem of such a type is the following. Let $G$ be a connected Lie group, let $R$ be its radical, let $S$ be a maximal connected semi-simple subgroup, and let $C$ be a maximal connected compact normal subgroup of $S$. Then the subgroup $RC$ has the Bieberbach property in $G$ . It is also known that the Bieberbach property is displayed by the nilpotent radical of a connected solvable Lie group  and by the commutator subgroup of a simply-connected nilpotent Lie group .

Topological methods (cf. Discrete group of transformations) can be used to prove that any uniform discrete subgroup of a connected Lie group is a finitely-presentable group . In fact, any lattice in a connected Lie group is finitely presentable , .