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...rst1=N.H. |last2=Ostaszewski |first2=A.J. |title=Normed versus topological groups: Dichotomy and duality |journal=Dissertationes Mathematicae |date=2010 |vol | title = A characterization of free abelian groups ...

...76}}.<ref>{{Citation | last1=Jørgensen | first1=Troels | title=On discrete groups of Möbius transformations | jstor=2373814 | mr=0427627 | year=1976 | journa The inequality states that if ''A'' and ''B'' generate a non-elementary discrete subgroup of the SL₂('''C'''), then ...

...are used in the theory of [[Lattice (discrete subgroup)|lattices]] in Lie groups, often under the name ''field of definition''. == Fuchsian and Kleinian groups == ...

...s a subgroup of the rational points of ''G'', but need not be a [[lattice (discrete subgroup)|lattice]]: it may be a so-called [[thin group (algebraic group th ...linear group]] over the integers, and in more general classes of algebraic groups ''G'', is that the sequence of [[Cayley graph]]s for reductions Γ<sub>''p'' ...

...l Ludwig Siegel|editor-surname1= Annals of Mathematics|title=Discontinuous groups|volume=44|issue=4|at=pp.&nbsp;674−689|date=1943 [[Category:Topological groups]] ...

...-19-560525-9 | mr=0422505 | year=1975 | chapter=Some arithmetical discrete groups in Lobačevskiĭ spaces | pages=323–348}} [[Category:Reflection groups]] ...

...roups is essentially due to the book ''The Lie-Theory of Connected Pro-Lie Groups'' by [[Karl Heinrich Hofmann]] and [[Sidney Morris]], but has since attract ...groups, taken in the [[category theory#category|category]] of topological groups. ...

...ore article has been created. -->{{Not to be confused with|Direct limit of groups}} ...irst=Daniel |date=2009-07-26 |title=Limit groups for relatively hyperbolic groups. I. The basic tools |url=https://msp.org/agt/2009/9-3/p08.xhtml |journal=Al ...

...ents as its cells is also [[surjective function|surjective]]. Surjunctive groups were introduced by {{harvtxt|Gottschalk|1973}}. It is unknown whether every ...topology]] (called a [[prodiscrete topology]] because it is the product of discrete topologies). ...

'''Local rigidity''' theorems in the theory of discrete subgroups of [[Lie group]]s are results which show that small deformations ...irst1=André | author1-link=André Weil | title=On discrete subgroups of Lie groups. II | jstor=1970212 | mr=0137793 | year=1962 | journal=[[Annals of Mathemat ...

===<span id="higherram"></span>Higher ramification groups=== ...nite Galois extension, and that <math>v_K</math> is a [[Discrete valuation|discrete normalised valuation]] of ''K'', whose residue field has characteristic ''p ...

...umann dimension]]s of the resulting {{nowrap|<math>l^2</math>-cohomology}} groups, and computed several examples, which all turned out to be [[rational numbe | chapter = Elliptic operators, discrete groups and von Neumann algebras ...

...ssociated with a [[Polish space]]. Except in the case of [[discrete space|discrete]] Polish spaces, the standard Borel space is unique, up to [[isomorphism]] ...then the [[Borel σ-algebra]].<ref>Mackey, G.W. (1957): Borel structure in groups and their duals. [[Trans. Am. Math. Soc.]], 85, 134-165.</ref> ...

...thorlink=Charles Sims (mathematician) |title=Graphs and finite permutation groups |journal=[[Mathematische Zeitschrift]] |volume=95 |number=1 |year=1967 |pag ...Jan Saxl]], and [[Gary Seitz]] using the [[classification of finite simple groups]], in particular the fact that only finitely many [[isomorphism]] types of ...

...th> is said to be '''linearly topologized'''. If <math>A</math> is given a discrete topology, then <math>M</math> becomes a [[Topological module|topological <m ...h linear topology form a natural class of topological vector spaces ''over discrete fields'', analogous to the class of locally convex topological vector space ...

...can English|date=January 2019}}{{Short description|Equivalence relation of groups In [[mathematics]], specifically in [[group theory]], two groups are '''commensurable''' if they differ only by a finite amount, in a precis ...

...finite group is. It can be generalized to infinite [[group (mathematics)|groups]] equipped with a suitable [[probability measure]],<ref name=":0" /> and ca If one considers the [[discrete uniform distribution|uniform distribution]] on <math>G^2</math>, <math>p(G) ...

==Definition for finite groups== ...nski|first3=Grigori|title=Asymptotics of Plancherel measures for symmetric groups|journal=Journal of the American Mathematical Society|year=2000|series=13:49 ...

In mathematics, a '''convergence group''' or a '''discrete convergence group''' is a [[group (mathematics)|group]] <math>\Gamma</math> | title=Discrete quasiconformal groups I ...

...cs]], the '''Kazhdan–Margulis theorem''' is a statement asserting that a [[discrete subgroup]] in [[semisimple Lie group]]s cannot be too dense in the group. M ...</math> of the identity <math>e</math> in <math>G</math> such that for any discrete subgroup <math>\Gamma \subset G</math> there is an element <math>g \in G</m ...