Accola–Maclachlan bound

E904004

mathematical bound result in algebraic geometry result in the theory of Riemann surfaces

The Accola–Maclachlan bound is a refinement in algebraic geometry that gives an improved upper limit on the size of the automorphism group of a compact Riemann surface (or algebraic curve), sharpening the classical Hurwitz bound in certain cases.

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Statements (40)

Predicate	Object
instanceOf	mathematical bound ⓘ result in algebraic geometry ⓘ result in the theory of Riemann surfaces ⓘ
appliesAlsoTo	smooth projective algebraic curves over algebraically closed fields of characteristic 0 ⓘ
appliesTo	algebraic curves over the complex numbers ⓘ automorphism groups of compact Riemann surfaces ⓘ compact Riemann surfaces ⓘ
assumes	compactness of the Riemann surface ⓘ genus at least 2 ⓘ
comparesWith	Hurwitz bound NERFINISHED ⓘ
concerns	relationship between genus of a Riemann surface and size of its automorphism group ⓘ
context	finite group actions on Riemann surfaces ⓘ maximal automorphism groups of curves of given genus ⓘ
expresses	numerical constraint on \|Aut(X)\| in terms of the genus g of X ⓘ
field	Riemann surface theory NERFINISHED ⓘ algebraic geometry ⓘ automorphism groups of Riemann surfaces ⓘ complex analysis ⓘ group actions on Riemann surfaces ⓘ
gives	upper bound on the order of the automorphism group of a compact Riemann surface ⓘ
goal	sharpen the general upper bound on automorphism groups beyond Hurwitz’s 84(g−1) bound in special cases ⓘ
hasDomain	compact Riemann surfaces of genus g ≥ 2 ⓘ
historicalContext	20th century developments in the theory of Riemann surfaces ⓘ
improvesOn	classical Hurwitz bound in certain genera ⓘ
involves	finite groups of conformal automorphisms ⓘ hyperbolic geometry of Riemann surfaces ⓘ
language	algebro-geometric formulation ⓘ complex analytic formulation ⓘ
mathematicalObject	inequality relating genus and automorphism group order ⓘ
namedAfter	Colin Maclachlan NERFINISHED ⓘ Robert D. M. Accola NERFINISHED ⓘ
refines	Hurwitz bound NERFINISHED ⓘ
relatedTo	Fuchsian groups ⓘ Hurwitz surfaces NERFINISHED ⓘ automorphism groups of algebraic curves ⓘ uniformization of Riemann surfaces ⓘ
typeOf	group-theoretic bound in geometry ⓘ inequality in complex geometry ⓘ
usedFor	classifying Riemann surfaces with large automorphism groups ⓘ studying extremal Riemann surfaces with many automorphisms ⓘ

Referenced by (1)

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

Hurwitz bound on automorphism groups of curves → relatedResult → Accola–Maclachlan bound ⓘ