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...inian groups, the '''Riley slice''' of [[Schottky space]] is a family of [[Kleinian group]]s generated by two parabolic elements. generate a Kleinian group ''G'' with regular set Ω such that Ω/''G'' is a 4-times punctured sph ...

{{Short description|Inequality involving the traces of elements of a Kleinian group}} ...76}}.<ref>{{Citation | last1=Jørgensen | first1=Troels | title=On discrete groups of Möbius transformations | jstor=2373814 | mr=0427627 | year=1976 | journa ...

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

...eometry of the [[Teichmüller space]], of [[mapping class group]]s and of [[Kleinian group]]s. It was introduced by W.J.Harvey in 1978. ...is action allows to prove many interesting properties of the mapping class groups.{{sfn|Ivanov|1992|loc=Chapter 7}} ...

}}</ref> [[Kleinian group]]s,<ref>{{cite journal | title=Hausdorff dimension and Kleinian groups ...

...direct.com/science/article/pii/0021869372900580 ''Free subgroups in linear groups.''] [[Journal of Algebra]], vol. 20 (1972), pp. 250–270</ref> containing th ...1=''H'' = ''H''₁&thinsp;∗&thinsp;''H''₂}}. Since the groups {{math|''H''₁}} and {{math|''H''₂}} are infinite [[cy ...

==Cannon–Thurston maps and [[Kleinian group]]s== ===Kleinian representations of surface groups=== ...

...am Karrass, Donald Solitar, ''Combinatorial group theory. Presentations of groups in terms of generators and relations'', Reprint of the 1976 second edition, ...ng co-Hopfian is not a [[quasi-isometry]] invariant for finitely generated groups.<ref name=COR/> ...

...following work of [[Ralph Fox]], he was interested in morphisms to finite groups. Later on in Southampton, considering <math>\mathrm{SL}_2(\mathbb C)</math> ...last=Riley |first=Robert| title=Discrete parabolic representations of link groups |journal=[[Mathematika]]| volume=22 |issue=2 |year=1975a |pages=141–150 |do ...

...<math>M</math> in a way that generalizes the properties of the action of [[Kleinian group]] by [[Möbius transformation]]s on the ideal boundary <math>\mathbb S | title=Discrete quasiconformal groups I ...

...the notion of [[dimension of a vector space]]. Indeed, for [[p-group|''p''-groups]], the rank of the group ''P'' is the dimension of the vector space ''P''/Φ ...ich is automatically the case for dimensions of vector spaces, but not for groups such as [[affine group]]s. To distinguish these different definitions, one ...

...rbolic plane]] often exhibit particularly regular behaviour among Fuchsian groups and hyperbolic surfaces. === Arithmetic Fuchsian groups === ...

...tudying the [[Group action (mathematics)|action]] of [[group (mathematics)|groups]] on [[Real tree|'''R'''-trees]]. It was introduced in unpublished work of ...ina | first1=Mladen | last2=Feighn | first2=Mark | title=Stable actions of groups on real trees | doi=10.1007/BF01884300 | doi-access=free | mr=1346208 | yea ...

...erbolic geometry, [[minimal surface]] theory, [[hyperbolic group]]s, and [[Kleinian group]]s with their associated geometry, topology, and group theory. ...ichael|authorlink=Michael Kapovich|title=Hyperbolic Manifolds and Discrete Groups|year=2009|page=113|url=https://books.google.com/books?id=JRJ8VmfP-hcC&pg=PA ...

...geodesic extensions of an element). Cannon proved that a convex-cocompact Kleinian group has only finitely many cone types (with respect to a fixed finite gen ...ler|S. Hermiller]] and J. Meier, ''Measuring the tameness of almost convex groups''. [[Transactions of the American Mathematical Society]] vol. 353 (2001), n ...

Note that in general Lie groups this statement is far from being true; in particular, in a [[Nilpotent grou ...J. | title=Minimal co-volume hyperbolic lattices, II: Simple torsion in a Kleinian group | journal= Annals of Mathematics | volume=176 | date=2012 | pages=261 ...

...'''Du Val singularity''', also called '''simple surface singularity''', '''Kleinian singularity''', or '''rational double point''', is an isolated singularit ...

...rrete |first2=Juan Pablo |last3=Seade |first3=José |title=Complex Kleinian Groups |date=2013 |url=https://link.springer.com/book/10.1007/978-3-0348-0481-3 |l [[Category:Lie groups]] ...

...ym}}}} Doyle spirals have been used to study [[Kleinian group]]s, discrete groups of symmetries of [[hyperbolic space]], by embedding these spirals onto the | title = Spaces of Kleinian Groups ...

...NN extension]], via the notion of the '''fundamental group of a [[graph of groups]].''' Bass–Serre theory can be regarded as one-dimensional version of the [ ...rnal |doi=10.1016/0022-4049(93)90085-8|title=Covering theory for graphs of groups |year=1993 |last1=Bass |first1=Hyman |journal=Journal of Pure and Applied A ...