Ahlfors finiteness theorem

E898488

mathematical theorem theorem in Kleinian group theory theorem in complex analysis

The Ahlfors finiteness theorem is a fundamental result in the theory of Kleinian groups stating that, under suitable discreteness and analyticity conditions, the quotient of the domain of discontinuity has finite topological type.

Try in SPARQL Jump to: Statements Referenced by

Statements (42)

Predicate	Object
instanceOf	mathematical theorem ⓘ theorem in Kleinian group theory ⓘ theorem in complex analysis ⓘ
appliesTo	Kleinian groups NERFINISHED ⓘ discrete subgroups of PSL(2,C) ⓘ
assumes	group acts properly discontinuously on its domain of discontinuity ⓘ group is a discrete subgroup of Möbius transformations ⓘ group is finitely generated ⓘ
category	theorems about discrete groups of isometries ⓘ theorems in conformal dynamics ⓘ
concerns	domain of discontinuity of a Kleinian group ⓘ quotient of the domain of discontinuity by a Kleinian group ⓘ
concludes	each component of the quotient has finitely generated fundamental group ⓘ each component of the quotient has finitely many ends ⓘ quotient of the domain of discontinuity has finite topological type ⓘ quotient of the domain of discontinuity is a finite union of analytically finite Riemann surfaces ⓘ
context	action of Kleinian groups on the Riemann sphere ⓘ decomposition of the Riemann sphere into limit set and domain of discontinuity ⓘ
field	Kleinian groups ⓘ complex analysis ⓘ geometric function theory ⓘ hyperbolic geometry ⓘ low-dimensional topology ⓘ
generalizes	finiteness properties of Fuchsian groups ⓘ
hasConsequence	finiteness of moduli for certain Kleinian group quotients ⓘ structure theory of analytically finite Riemann surfaces ⓘ
implies	finiteness of conformal structure on quotient surfaces ⓘ finiteness of number of components of the quotient of the domain of discontinuity ⓘ
influenced	Sullivan’s work on Kleinian groups ⓘ modern 3-manifold theory ⓘ
namedAfter	Lars Ahlfors NERFINISHED ⓘ
provedBy	Lars Ahlfors NERFINISHED ⓘ
relatesTo	Riemann surfaces NERFINISHED ⓘ Teichmüller theory NERFINISHED ⓘ analytically finite Riemann surfaces ⓘ conformal structures ⓘ
timePeriod	mid 20th century ⓘ
typicalConclusion	quotient of domain of discontinuity is a finite-type Riemann surface or orbifold ⓘ
typicalHypothesis	Kleinian group is finitely generated NERFINISHED ⓘ
usedIn	classification of Kleinian groups ⓘ proofs of tameness-type results for Kleinian groups ⓘ study of hyperbolic 3-manifolds ⓘ

Referenced by (1)

Full triples — surface form annotated when it differs from this entity's canonical label.

Kleinian group → relatedTo → Ahlfors finiteness theorem ⓘ