Hurwitz's Theorem On Automorphisms
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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, Hurwitz's automorphisms theorem bounds the order of the group of
automorphism In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms ...
s, via orientation-preserving conformal mappings, of a compact Riemann surface of genus ''g'' > 1, stating that the number of such automorphisms cannot exceed 84(''g'' − 1). A group for which the maximum is achieved is called a Hurwitz group, and the corresponding Riemann surface a Hurwitz surface. Because compact Riemann surfaces are synonymous with non-singular complex projective algebraic curves, a Hurwitz surface can also be called a Hurwitz curve.Technically speaking, there is an equivalence of categories between the category of compact Riemann surfaces with the orientation-preserving conformal maps and the category of non-singular complex projective algebraic curves with the algebraic morphisms. The theorem is named after
Adolf Hurwitz Adolf Hurwitz (; 26 March 1859 – 18 November 1919) was a German mathematician who worked on algebra, analysis, geometry and number theory. Early life He was born in Hildesheim, then part of the Kingdom of Hanover, to a Jewish family and died ...
, who proved it in . Hurwitz's bound also holds for algebraic curves over a field of characteristic 0, and over fields of positive characteristic ''p''>0 for groups whose order is coprime to ''p'', but can fail over fields of positive characteristic ''p''>0 when ''p'' divides the group order. For example, the double cover of the projective line ''y''2 = ''xp'' −''x'' branched at all points defined over the prime field has genus ''g''=(''p''−1)/2 but is acted on by the group SL2(''p'') of order ''p''3−''p''.


Interpretation in terms of hyperbolicity

One of the fundamental themes in
differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...
is a trichotomy between the
Riemannian manifold In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real manifold, real, smooth manifold ''M'' equipped with a positive-definite Inner product space, inner product ...
s of positive, zero, and negative
curvature In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane. For curves, the canonic ...
''K''. It manifests itself in many diverse situations and on several levels. In the context of compact Riemann surfaces ''X'', via the Riemann uniformization theorem, this can be seen as a distinction between the surfaces of different topologies: * ''X'' a sphere, a compact Riemann surface of genus zero with ''K'' > 0; * ''X'' a flat torus, or an elliptic curve, a Riemann surface of genus one with ''K'' = 0; * and ''X'' a
hyperbolic surface In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P ...
, which has genus greater than one and ''K'' < 0. While in the first two cases the surface ''X'' admits infinitely many conformal automorphisms (in fact, the conformal automorphism group is a complex
Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
of dimension three for a sphere and of dimension one for a torus), a hyperbolic Riemann surface only admits a discrete set of automorphisms. Hurwitz's theorem claims that in fact more is true: it provides a uniform bound on the order of the automorphism group as a function of the genus and characterizes those Riemann surfaces for which the bound is sharp.


Statement and proof

Theorem: Let X be a smooth connected Riemann surface of genus g \ge 2. Then its automorphism group \mathrm(X) has size at most 84(g-1). ''Proof:'' Assume for now that G = \mathrm(X) is finite (we'll prove this at the end). * Consider the quotient map X \to X/G. Since G acts by holomorphic functions, the quotient is locally of the form z \to z^n and the quotient X/G is a smooth Riemann surface. The quotient map X \to X/G is a branched cover, and we will see below that the ramification points correspond to the orbits that have a non trivial stabiliser. Let g_0 be the genus of X/G. * By the Riemann-Hurwitz formula, 2g-2 \ = \ , G, \cdot \left( 2g_0-2 + \sum_^k \left(1-\frac\right)\right) where the sum is over the k ramification points p_i \in X/G for the quotient map X \to X/G. The ramification index e_i at p_i is just the order of the stabiliser group, since e_i f_i = \deg(X/\, X/G) where f_i the number of pre-images of p_i (the number of points in the orbit), and \deg(X/\, X/G) = , G, . By definition of ramification points, e_i \ge 2 for all k ramification indices. Now call the righthand side , G, R and since g \ge 2 we must have R>0. Rearranging the equation we find: * If g_0 \ge 2 then R \ge 2, and , G, \le (g-1) * If g_0 = 1 , then k \ge 1 and R\ge 0 + 1 - 1/2 = 1/2 so that , G, \le 4(g-1), * If g_0 = 0, then k \ge 3 and ** if k \ge 5 then R \ge -2 + k(1 - 1/2) \ge 1/2, so that , G, \le 4(g-1) ** if k=4 then R \ge -2 + 4 - 1/2 - 1/2 - 1/2 - 1/3 = 1/6, so that , G, \le 12(g-1), ** if k=3 then write e_1 = p,\, e_2 = q, \, e_3 = r. We may assume 2 \le p\le q\ \le r. *** if p \ge 3 then R \ge -2 + 3 - 1/3 - 1/3 - 1/4 = 1/12 so that , G, \le 24(g-1), *** if p = 2 then **** if q \ge 4 then R \ge -2 + 3 - 1/2 - 1/4 - 1/5 = 1/20 so that , G, \le 40(g-1), **** if q = 3 then R \ge -2 + 3 - 1/2 - 1/3 - 1/7 = 1/42 so that , G, \le 84(g-1). In conclusion, , G, \le 84(g-1). To show that G is finite, note that G acts on the cohomology H^*(X,\mathbf) preserving the Hodge decomposition and the lattice H^1(X,\mathbf). *In particular, its action on V=H^(X,\mathbf) gives a homomorphism h: G \to \mathrm(V) with discrete image h(G). *In addition, the image h(G) preserves the natural non degenerate Hermitian inner product (\omega,\eta)= i \int\bar\wedge\eta on V. In particular the image h(G) is contained in the unitary group \mathrm(V) \subset \mathrm(V) which is compact. Thus the image h(G) is not just discrete, but finite. * It remains to prove that h: G \to \mathrm(V) has finite kernel. In fact, we will prove h is injective. Assume \phi \in G acts as the identity on V. If \mathrm(\phi) is finite, then by the Lefschetz fixed-point theorem, , \mathrm(\phi), = 1 - 2\mathrm(h(\phi)) + 1 = 2 - 2\mathrm(\mathrm_V) = 2 - 2g < 0. This is a contradiction, and so \mathrm(\phi) is infinite. Since \mathrm(\phi) is a closed complex sub variety of positive dimension and X is a smooth connected curve (i.e. \dim_(X) = 1), we must have \mathrm(\phi) = X. Thus \phi is the identity, and we conclude that h is injective and G \cong h(G) is finite. Q.E.D. Corollary of the proof: A Riemann surface X of genus g \ge 2 has 84(g-1) automorphisms if and only if X is a branched cover X \to \mathbf^1 with three ramification points, of indices ''2'',''3'' and ''7''.


The idea of another proof and construction of the Hurwitz surfaces

By the uniformization theorem, any hyperbolic surface ''X'' – i.e., the Gaussian curvature of ''X'' is equal to negative one at every point – is
covered Cover or covers may refer to: Packaging * Another name for a lid * Cover (philately), generic term for envelope or package * Album cover, the front of the packaging * Book cover or magazine cover ** Book design ** Back cover copy, part of ...
by the hyperbolic plane. The conformal mappings of the surface correspond to orientation-preserving automorphisms of the hyperbolic plane. By the Gauss–Bonnet theorem, the area of the surface is : A(''X'') = − 2π χ(''X'') = 4π(''g'' − 1). In order to make the automorphism group ''G'' of ''X'' as large as possible, we want the area of its
fundamental domain Given a topological space and a group acting on it, the images of a single point under the group action form an orbit of the action. A fundamental domain or fundamental region is a subset of the space which contains exactly one point from each o ...
''D'' for this action to be as small as possible. If the fundamental domain is a triangle with the vertex angles π/p, π/q and π/r, defining a tiling of the hyperbolic plane, then ''p'', ''q'', and ''r'' are integers greater than one, and the area is : A(''D'') = π(1 − 1/''p'' − 1/''q'' − 1/''r''). Thus we are asking for integers which make the expression :1 − 1/''p'' − 1/''q'' − 1/''r'' strictly positive and as small as possible. This minimal value is 1/42, and :1 − 1/2 − 1/3 − 1/7 = 1/42 gives a unique triple of such integers. This would indicate that the order , ''G'', of the automorphism group is bounded by : A(''X'')/A(''D'')  ≤  168(''g'' − 1). However, a more delicate reasoning shows that this is an overestimate by the factor of two, because the group ''G'' can contain orientation-reversing transformations. For the orientation-preserving conformal automorphisms the bound is 84(''g'' − 1).


Construction

To obtain an example of a Hurwitz group, let us start with a (2,3,7)-tiling of the hyperbolic plane. Its full symmetry group is the full
(2,3,7) triangle group In the theory of Riemann surfaces and hyperbolic geometry, the triangle group (2,3,7) is particularly important. This importance stems from its connection to Hurwitz surfaces, namely Riemann surfaces of genus ''g'' with the largest possible order, ...
generated by the reflections across the sides of a single fundamental triangle with the angles π/2, π/3 and π/7. Since a reflection flips the triangle and changes the orientation, we can join the triangles in pairs and obtain an orientation-preserving tiling polygon. A Hurwitz surface is obtained by 'closing up' a part of this infinite tiling of the hyperbolic plane to a compact Riemann surface of genus ''g''. This will necessarily involve exactly 84(''g'' − 1) double triangle tiles. The following two regular tilings have the desired symmetry group; the rotational group corresponds to rotation about an edge, a vertex, and a face, while the full symmetry group would also include a reflection. The polygons in the tiling are not fundamental domains – the tiling by (2,3,7) triangles refines both of these and is not regular. Wythoff constructions yields further uniform tilings, yielding eight uniform tilings, including the two regular ones given here. These all descend to Hurwitz surfaces, yielding tilings of the surfaces (triangulation, tiling by heptagons, etc.). From the arguments above it can be inferred that a Hurwitz group ''G'' is characterized by the property that it is a finite quotient of the group with two generators ''a'' and ''b'' and three relations :a^2 = b^3 = (ab)^7 = 1, thus ''G'' is a finite group generated by two elements of orders two and three, whose product is of order seven. More precisely, any Hurwitz surface, that is, a hyperbolic surface that realizes the maximum order of the automorphism group for the surfaces of a given genus, can be obtained by the construction given. This is the last part of the theorem of Hurwitz.


Examples of Hurwitz groups and surfaces

The smallest Hurwitz group is the projective special linear group PSL(2,7), of order 168, and the corresponding curve is the Klein quartic curve. This group is also isomorphic to
PSL(3,2) In mathematics, the projective special linear group , isomorphic to , is a finite simple group that has important applications in algebra, geometry, and number theory. It is the automorphism group of the Klein quartic as well as the symmetry gr ...
. Next is the Macbeath curve, with automorphism group PSL(2,8) of order 504. Many more finite simple groups are Hurwitz groups; for instance all but 64 of the alternating groups are Hurwitz groups, the largest non-Hurwitz example being of degree 167. The smallest alternating group that is a Hurwitz group is A15. Most projective special linear groups of large rank are Hurwitz groups, . For lower ranks, fewer such groups are Hurwitz. For ''n''''p'' the order of ''p'' modulo 7, one has that PSL(2,''q'') is Hurwitz if and only if either ''q''=7 or ''q'' = ''p''''n''''p''. Indeed, PSL(3,''q'') is Hurwitz if and only if ''q'' = 2, PSL(4,''q'') is never Hurwitz, and PSL(5,''q'') is Hurwitz if and only if ''q'' = 74 or ''q'' = ''p''''n''''p'', . Similarly, many groups of Lie type are Hurwitz. The finite classical groups of large rank are Hurwitz, . The
exceptional Lie group In mathematics, a simple Lie group is a connected non-abelian Lie group ''G'' which does not have nontrivial connected normal subgroups. The list of simple Lie groups can be used to read off the list of simple Lie algebras and Riemannian symm ...
s of type G2 and the Ree groups of type 2G2 are nearly always Hurwitz, . Other families of exceptional and twisted Lie groups of low rank are shown to be Hurwitz in . There are 12 sporadic groups that can be generated as Hurwitz groups: the Janko groups J1, J2 and J4, the
Fischer group In the area of modern algebra known as group theory, the Fischer groups are the three sporadic simple groups Fi22, Fi23 and Fi24 introduced by . 3-transposition groups The Fischer groups are named after Bernd Fischer who discovered them ...
s Fi22 and Fi'24, the Rudvalis group, the Held group, the Thompson group, the Harada–Norton group, the third Conway group Co3, the Lyons group, and the
Monster A monster is a type of fictional creature found in horror, fantasy, science fiction, folklore, mythology and religion. Monsters are very often depicted as dangerous and aggressive with a strange, grotesque appearance that causes terror and fe ...
, .


Automorphism groups in low genus

The largest , Aut(''X''), can get for a Riemann surface ''X'' of genus ''g'' is shown below, for 2≤''g''≤10, along with a surface ''X''0 with , Aut(''X''0), maximal. In this range, there only exists a Hurwitz curve in genus ''g''=3 and ''g''=7.


Generalizations

The concept of a Hurwitz surface can be generalized in several ways to a definition that has examples in all but a few genera. Perhaps the most natural is a "maximally symmetric" surface: One that cannot be continuously modified through equally symmetric surfaces to a surface whose symmetry properly contains that of the original suface. This is possible for all orientable compact genera (see above section "Automorphism groups in low genus").


See also

*
(2,3,7) triangle group In the theory of Riemann surfaces and hyperbolic geometry, the triangle group (2,3,7) is particularly important. This importance stems from its connection to Hurwitz surfaces, namely Riemann surfaces of genus ''g'' with the largest possible order, ...


Notes


References

* * * * * * * * {{Algebraic curves navbox Theorems in algebraic geometry Riemann surfaces Theorems in group theory Theorems in complex geometry