Discrete group

Discrete group
Original: Jakob.scholbach Vector: Pbroks13 · CC BY-SA 3.0 · source
Name	Discrete group
Type	Algebraic/topological structure
Related	Group, Topology, Lie group, Fundamental group, Covering space, Fuchsian group

Discrete group

A discrete group is a group equipped with the discrete topology or a subgroup that is discrete in a topological ambient space; it appears across Felix Klein's Erlangen program, Henri Poincaré's work on automorphic functions, and modern studies in William Thurston's geometry. Discrete groups connect algebraic structures like Galois theory and Abelian groups with geometric and analytic objects studied by Bernhard Riemann, David Hilbert, and researchers in Élie Cartan's theory of Lie groups. They underlie examples in Hyperbolic geometry, Teichmüller theory, and the theory of modular forms investigated by Srinivasa Ramanujan and Goro Shimura.

Definition and basic properties

A group G with the discrete topology is one for which every singleton {g} is open; this notion appears in works of André Weil and Hermann Weyl in harmonic analysis and representation theory. In a topological group context, a subgroup H of a topological group K is discrete if the subspace topology on H is discrete; this condition features in studies by Harish-Chandra and George Mackey on unitary representations and by Gregory Margulis in rigidity theory. Basic properties include that discrete subgroups are closed and locally compact when considered inside locally compact ambient groups such as SL(2,R), GL(n,R), or isometry groups of hyperbolic space. Discreteness interacts with finiteness conditions studied by Max Dehn and Issai Schur and with separation axioms discussed by Maurice Fréchet.

Examples and classes

Classical examples include lattices in Euclidean space generated by translations such as the integer lattice Z^n studied since Carl Friedrich Gauss; Fuchsian groups like the Modular group PSL(2,Z) arise in the work of Henri Poincaré and Ernst Hecke. Discrete subgroups of SL(2,C) produce Kleinian groups central to Ahlfors and Maskit's investigations and to William Thurston's hyperbolic 3-manifold theory. Crystallographic groups classified by Evgraf Fedorov and Arthur Moritz Schönflies give examples tied to Bravais lattices and International Tables for Crystallography. Other classes include arithmetic groups studied by Goro Shimura and Armand Borel, Coxeter groups analyzed by H.S.M. Coxeter, and free groups explored by Jakob Nielsen and Walter von Dyck.

Topological and metric discreteness

Metric discreteness appears in the context of Gromov hyperbolic spaces and coarse geometry developed by Mikhail Gromov; it is crucial for proper discontinuity in actions considered by William Thurston and John Milnor. Topological discreteness is essential in covering space theory originating with Hermann Weyl and Poincaré; discrete subgroups of Fundamental groups determine deck transformation groups investigated by L.E.J. Brouwer and H. Hopf. In geometric group theory, quasi-isometry invariants introduced by Mikhail Gromov and explored by Cornelia Druţu and Mark Sapir distinguish discrete metric properties of groups such as growth rates first studied by Émile Borel and Harald Bohr.

Discrete subgroups of Lie groups

Discrete subgroups of Lie groups like SL(2,R), SO(n,1), and GL(n,R) are central to Harmonic analysis and representation theory advanced by Harish-Chandra and George Mackey. Lattices in semisimple Lie groups were classified in part by G. A. Margulis's arithmeticity and superrigidity theorems and in earlier work by Armand Borel and Harish-Chandra. The theory of discrete series representations initiated by Harish-Chandra links discrete subgroups to automorphic forms studied by Atle Selberg and Robert Langlands. Discreteness criteria such as the Selberg lemma and Mostow rigidity originate in works of Atle Selberg and G. D. Mostow.

Actions and geometric applications

Properly discontinuous actions of discrete groups on manifolds produce quotient spaces central to Thurston's geometrization conjecture and to the construction of locally symmetric spaces exemplified by Armand Borel and Hermann Weyl. Fuchsian and Kleinian group actions give rise to Riemann surfaces and hyperbolic 3-manifolds studied by Bers, Ahlfors, Maskit, and William Thurston. Discrete isometry groups of CAT(0) spaces and buildings feature in the work of Jacques Tits and Mikhail Gromov; they underpin the study of Bass–Serre theory by Hyman Bass and Jean-Pierre Serre on groups acting on trees. Applications extend to the study of modular curves of André Weil and Serge Lang and to arithmetic manifolds investigated by Goro Shimura and Don Zagier.

Algebraic and combinatorial aspects

Algebraic properties of discrete groups include residual finiteness studied by Gustav Magnus and Malcev and subgroup growth investigated by Grigorchuk and Alexander Lubotzky. Combinatorial group theory from Max Dehn to John Stallings and Roger Lyndon analyzes presentations, the word problem, and small cancellation conditions. Coxeter groups and Artin groups classified by H.S.M. Coxeter and E. Artin demonstrate combinatorial structure alongside geometric actions explored by Bridson and Haefliger. Growth functions and amenability connect to work of John von Neumann and Mikhail Gromov.

Historical notes and important results

Historical development traces from Évariste Galois and lattice studies by Carl Friedrich Gauss, through Poincaré's uniformization and Kleinian group theory, to 20th-century advances by Harish-Chandra, Atle Selberg, and G. A. Margulis. Landmark results include Margulis's arithmeticity and superrigidity, Mostow rigidity by G. D. Mostow, the Selberg lemma, and the Borel–Harish-Chandra theorem. Progress in geometric group theory by Mikhail Gromov, and rigidity and counting results by G. A. Margulis and Elon Lindenstrauss continue to shape modern research influenced by figures such as William Thurston, Maryam Mirzakhani, and Curtis T. McMullen.

Category:Group theory