Hurwitz surfaces
E907201
Hurwitz surfaces are compact Riemann surfaces of maximal possible automorphism group size for their genus, achieving the Hurwitz bound of 84(g−1) symmetries.
Observed surface forms (4)
| Surface form | Occurrences |
|---|---|
| Hurwitz surface | 0 |
| Klein quartic | 0 |
| Macbeath surface | 0 |
| Fricke–Macbeath curve | 0 |
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
Hurwitz surface
ⓘ
Hurwitz surface ⓘ Hurwitz surface ⓘ algebraic curve ⓘ compact Riemann surface ⓘ complex algebraic curve ⓘ geometric object ⓘ |
| characterizedBy | |Aut(X)| = 84(g(X)−1) ⓘ |
| correspondsTo | points with maximal automorphism group in moduli space of curves ⓘ |
| definedOver | complex numbers ⓘ |
| hasAutomorphismGroup |
Hurwitz group
NERFINISHED
ⓘ
PSL(2,7) NERFINISHED ⓘ finite group ⓘ group of conformal automorphisms ⓘ |
| hasAutomorphismGroupOrder |
168
ⓘ
504 ⓘ 84(g−1) ⓘ |
| hasAutomorphismGroupOrderUpperBound | 84(g−1) ⓘ |
| hasBelyiMap | ramified over three points with ramification type (2,3,7) ⓘ |
| hasCurvature | negative ⓘ |
| hasEulerCharacteristic | 2−2g ⓘ |
| hasExample |
Fricke–Macbeath curve
NERFINISHED
ⓘ
Klein quartic NERFINISHED ⓘ Macbeath surface NERFINISHED ⓘ |
| hasFundamentalGroup | torsion-free subgroup of (2,3,7) triangle group ⓘ |
| hasGenus |
3
ⓘ
7 ⓘ g ≥ 2 ⓘ |
| hasMetric | hyperbolic metric ⓘ |
| hasProperty | maximal automorphism group for its genus ⓘ |
| hasSymmetryType | maximally symmetric for its genus ⓘ |
| is | quotient of hyperbolic plane by a torsion-free subgroup of (2,3,7) triangle group ⓘ |
| isQuotientOf | hyperbolic plane ⓘ |
| liesIn | moduli space of curves of genus g ⓘ |
| maximizes | number of conformal automorphisms for given genus ⓘ |
| namedAfter | Adolf Hurwitz NERFINISHED ⓘ |
| relatedTo |
Fuchsian group
ⓘ
Galois Belyi map ⓘ Hurwitz group NERFINISHED ⓘ Hurwitz’s automorphism theorem NERFINISHED ⓘ triangle group (2,3,7) NERFINISHED ⓘ |
| satisfies | Hurwitz bound ⓘ |
| studiedIn |
Riemann surface theory
ⓘ
algebraic geometry ⓘ complex analysis ⓘ geometric group theory ⓘ |
| uniformizedBy | (2,3,7) triangle group NERFINISHED ⓘ |
| usedIn | construction of large automorphism groups of curves ⓘ |
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.