In
mathematics, Hurwitz's automorphisms theorem bounds the order of the group of
automorphisms, via
orientation-preserving conformal mapping
In mathematics, a conformal map is a function that locally preserves angles, but not necessarily lengths.
More formally, let U and V be open subsets of \mathbb^n. A function f:U\to V is called conformal (or angle-preserving) at a point u_0\i ...
s, of a compact
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed ve ...
of
genus
Genus ( plural genera ) is a taxonomic rank used in the biological classification of living and fossil organisms as well as viruses. In the hierarchy of biological classification, genus comes above species and below family. In binomial n ...
''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
In category theory, a branch of abstract mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are "essentially the same". There are numerous examples of categorical equivalences ...
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, 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 = ''x
p'' −''x'' branched at all points defined over the prime field has genus ''g''=(''p''−1)/2 but is acted on by the group SL
2(''p'') of order ''p''
3−''p''.
Interpretation in terms of hyperbolicity
One of the fundamental themes in
differential geometry 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, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent spac ...
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 can ...
''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 sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
, a compact Riemann surface of
genus
Genus ( plural genera ) is a taxonomic rank used in the biological classification of living and fossil organisms as well as viruses. In the hierarchy of biological classification, genus comes above species and below family. In binomial n ...
zero with ''K'' > 0;
* ''X'' a flat
torus
In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle.
If the axis of revolution does not ...
, 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 addit ...
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
be a smooth connected Riemann surface of genus
. Then its automorphism group
has size at most
.
''Proof:'' Assume for now that
is finite (we'll prove this at the end).
* Consider the quotient map
. Since
acts by holomorphic functions, the quotient is locally of the form
and the quotient
is a smooth Riemann surface. The quotient map
is a branched cover, and we will see below that the ramification points correspond to the orbits that have a non trivial stabiliser. Let
be the genus of
.
* By the
Riemann-Hurwitz formula,
where the sum is over the
ramification points
for the quotient map
. The ramification index
at
is just the order of the stabiliser group, since
where
the number of pre-images of
(the number of points in the orbit), and
. By definition of ramification points,
for all
ramification indices.
Now call the righthand side
and since
we must have
. Rearranging the equation we find:
* If
then
, and
* If
, then
and
so that
,
* If
, then
and
** if
then
, so that
** if
then
, so that
,
** if
then write
. We may assume
.
*** if
then
so that
,
*** if
then
**** if
then
so that
,
**** if
then
so that
.
In conclusion,
.
To show that
is finite, note that
acts on the
cohomology
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewe ...
preserving the
Hodge decomposition and the
lattice .
*In particular, its action on
gives a homomorphism
with
discrete
Discrete may refer to:
*Discrete particle or quantum in physics, for example in quantum theory
*Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit
*Discrete group, a ...
image
.
*In addition, the image
preserves the natural non degenerate
Hermitian inner product on
. In particular the image
is contained in the
unitary group
In mathematics, the unitary group of degree ''n'', denoted U(''n''), is the group of unitary matrices, with the group operation of matrix multiplication. The unitary group is a subgroup of the general linear group . Hyperorthogonal group i ...
which is
compact. Thus the image
is not just discrete, but finite.
* It remains to prove that
has finite kernel. In fact, we will prove
is injective. Assume
acts as the identity on
. If
is finite, then by the
Lefschetz fixed-point theorem,
This is a contradiction, and so
is infinite. Since
is a closed complex sub variety of positive dimension and
is a smooth connected curve (i.e.
), we must have
. Thus
is the identity, and we conclude that
is injective and
is finite.
Q.E.D.
Corollary of the proof: A Riemann surface
of genus
has
automorphisms if and only if
is a branched cover
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 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 construction
In geometry, a Wythoff construction, named after mathematician Willem Abraham Wythoff, is a method for constructing a uniform polyhedron or plane tiling. It is often referred to as Wythoff's kaleidoscopic construction.
Construction process
...
s 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
:
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 group
In mathematics, an alternating group is the group of even permutations of a finite set. The alternating group on a set of elements is called the alternating group of degree , or the alternating group on letters and denoted by or
Basic pr ...
s are Hurwitz groups, the largest non-Hurwitz example being of degree 167. The smallest alternating group that is a Hurwitz group is A
15.
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'' = 7
4 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 groups 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 J
1, J
2 and J
4, 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 Fi
22 and Fi'
24, the
Rudvalis group, the
Held group, the
Thompson group, the
Harada–Norton group, the third
Conway group Co
3, 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
Notes
References
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