Discrete group of transformations

A group of homeomorphisms of a Hausdorff space that satisfies the following condition: It is possible to find neighbourhoods of arbitrary points such that the set

is finite. The stabilizer

of a point with respect to a discrete group of transformations is finite, while the orbit of an arbitrary point is discrete. If is a metric space and the transformations from are isometries, these two conditions are sufficient for to be a discrete group of transformations.

Examples.

1) The group of parallel translations of the real plane over all possible integer vectors:

2) Let be the complex upper half-plane

considered with the ordinary Hausdorff topology, and let be the group of Möbius transformations of the form

where , , , and are integers and (the Kleinian modular group).

3) Any finite group of homeomorphisms of a Hausdorff space . (The example of an irreducible algebraic variety with the Zariski topology shows that the condition of separability of is essential.)

4) The group of covering transformations of an arbitrary regular covering , where is connected and locally path-connected, while is a Hausdorff space, is a freely-acting (i.e. for any ) discrete group of transformations; the covering itself coincides with the mapping of factorization by this group. Conversely, if is a freely-acting discrete group of transformations of a connected topological space , then the quotient space is a Hausdorff space, and the quotient mapping is a regular covering of with as group of covering transformations. In particular, by virtue of the Poincaré–Koebe uniformization theorem, any Riemann surface, apart from a few trivial exceptions, may be obtained by factorization of the complex upper half-plane by a freely-acting discrete group of Möbius transformations with real coefficients (a so-called Fuchsian group).

5) In the theory of moduli of Riemann surfaces (and, more generally, of moduli of complex manifolds of some given type), discrete groups of transformations appear as modular groups (cf. Modular group). The simplest such group is discussed in Example 2.

6) Discrete groups of transformations include the crystallographic groups (cf. Crystallographic group). A fairly wide class of discrete groups of transformations, which includes Fuchsian and crystallographic groups, is constituted by discrete subgroups (cf. Discrete subgroup) of topological groups (in particular, of Lie groups), considered as groups of transformations of homogeneous spaces.

A closed subset of a topological space with a discrete group of transformations is called a fundamental domain of the group if it is the closure of an open subset and if the sets where have pairwise no common interior points and form a locally finite covering of . For instance, for the group of parallel translations in Example 1, the square

(*)

may be taken as a fundamental domain; the same purpose will be served by any parallelogram with vertices at integral points which has no integral points inside or on the sides, while in the case of a Kleinian modular group (Example 2) one may take the so-called modular figure

In many cases a fundamental domain can be constructed. For instance, if is a complete Riemannian manifold, if is the discrete group of transformations of consisting of the isometries of this space, and if is some point for which the stabilizer is trivial, then the Dirichlet domain

may be taken as a fundamental domain. In the above formula denotes the distance between two points and from . If is a simply-connected complete space of constant curvature, i.e. a sphere, a Euclidean space or a Lobachevskii space, a Dirichlet domain is a convex polyhedron.

The construction of a fundamental domain and the study of its properties furnish important information about the discrete group of transformations. Thus, the quotient space is obtained from a fundamental domain by way of "glueing" certain boundary points. For example, the quotient space of the group of parallel translations (Example 1) is obtained from the square (*) by glueing the opposite sides and is homeomorphic to a two-dimensional torus. The concept of a fundamental domain forms the base of the combinatorial-geometric method in the theory of discrete groups of transformations which appears in the studies of H. Poincaré on Fuchsian [1] and Kleinian [2] groups. The method makes it possible, on one hand, to clarify the structure of a discrete group of transformations as an abstract group (i.e. to find its generators and defining relations) and, on the other hand, to prove the discreteness and to find a fundamental domain of a group of transformations with given generators. The principle of this method is as follows. Let be a discrete group of isometries of an -dimensional simply-connected complete space of constant curvature, and let be a convex polyhedron which is a fundamental domain. The group is then generated by the set

All possible relations of the following two types may be taken as defining relations: where , and where ,

if , and if [7], [3], [6]. Conversely, let be a convex polyhedron in an -dimensional simply-connected complete space of constant curvature (including the degenerate case in which certain bihedral angles of the polyhedron are equal to ), and let an isometry of such that be given for each -dimensional face of the polyhedron . Also, 1) let there exist a face such that for each -dimensional face of ; and 2) let, for each -dimensional face of there exist a sequence of -dimensional faces of such that ,

and such that the polyhedra , have pairwise no common interior points. Under these conditions the group of isometries of generated by the transformations is discrete and the polyhedron is a fundamental domain. This is a consequence of a more general result obtained by A.D. Aleksandrov [4] concerning the filling of a space by convex polyhedra (see also [8]). The following description of freely-acting Fuchsian groups with a compact quotient space, which is due to Poincaré, may serve as an example of the above-said. In this context, the complex upper half-plane is taken to be the standard model of the Lobachevskii geometry (Poincaré's model of the Lobachevskii plane). As fundamental domain of any Fuchsian group of the type dealt with here one may take a convex bounded -gon having the following properties: a) the sum of its interior angles is ; and b) if, for a given direction of traversal of the boundary of the polygon , one denotes its sides by , , , , , then the length of will equal the length of for all . The figure shows such a Dirichlet domain for .

Figure: d033080a

If one now denotes by , , the isometries of the plane which preserve orientation and map, with a change of the direction, to if is even, and to if is odd (it is assumed that the directions of the sides of are those induced by the direction of traversal of ), the set is a system of generators for . The unique relation between these generators has the form

Conversely, if is an arbitrary convex bounded polygon which satisfies the conditions a) and b), then the group generated by the isometries is a freely-acting Fuchsian group, while the compact manifold is a compact Riemann surface of genus .

The cohomology theory of discrete groups of transformations consists of studying the connection between the cohomology of the space , of the space and of the group . In particular (Example 4), if is a discrete group of transformations which is the group of covering transformations of a regular covering , where is an acyclic topological space (i.e. if and ), then the singular cohomology of and the cohomology of as an abstract group with coefficients in an Abelian group (with the trivial structure of a -module) are connected by certain isomorphisms:

which are natural with respect to [10]. In the general case the connection between the above cohomology groups is expressed with the aid of certain spectral sequences [9], [10].

See also Automorphic form; Automorphic function; Arithmetic group.

References

[1]	H. Poincaré, "Théorie des groupes fuchsiennes" , Oeuvres , 2 , Gauthier-Villars (1952) pp. 108–168 (Acta Math. (1982), 1–62)
[2]	H. Poincaré, "Mémoire sur les groupes kleinéens" , Oeuvres , 2 , Gauthier-Villars (1952) pp. 258–299 (Acta Math. (1883), 49–92)
[3]	M. Gerstenhaber, "On the algebraic structure of discontinuous groups" Proc. Amer. Math. Soc. , 4 (1953) pp. 745–750
[4]	A.D. Aleksandrov, "On a completion of a space of polyhedra" Vestnik Leningrad. Gos. Univ. , 9 : 2 (1954) pp. 34–43 (In Russian)
[5]	H.S.M. Coxeter, W.O.J. Moser, "Generators and relations for discrete groups" , Springer (1972)
[6]	A. Weil, "Discrete subgroups of Lie groups" Ann. of Math. , 72 (1960) pp. 369–384
[7]	A.M. Macbeath, "Groups of homeomorphisms of a simply connected space" Ann. of Math. , 79 (1964) pp. 473–488
[8]	H. Abels, "Geometrische Erzeugung von diskontinuierlichen Gruppen" , Univ. Münster (1966)
[9]	A. Grothendieck, "Sur quelques points d'algèbre homologique" Tohoku Math. J. , 9 (1957) pp. 119–221
[10]	S. MacLane, "Homology" , Springer (1963)
[11]	J. Lehner, "Discontinuous groups and automorphic functions" , Amer. Math. Soc. (1964)
[12]	J.P. Serre, "Cohomogie des groupes discretes" C.R. Acad. Sci. Paris , 268 (1969) pp. 268–271

References

[a1]	A.F. Baerdon, "The geometry of discrete groups" , Springer (1983)
[a2]	J.A. Wolf, "Spaces of constant curvature" , McGraw-Hill (1967)
[a3]	A. Borel, N. Wallach, "Continuous cohomology, discrete subgroups and representations of reductive groups" , Princeton Univ. Press (1980)