Hurwitz theorem automorphism pdf

Hurwitzs theorem richard koch february 19, 2015 theorem 1 hurwitz. Generalization of a theorem of hurwitz queens university. An automorphism of a graph is a permutation of its vertex set that preserves incidences of vertices and edges. Any curve with 84gr 1 automorphisms must be uniformized by a normal subgroup of the triangle group 2, 3, 7, and, by a closer analysis of possible finite. For instance, in genus 3, the hurwitz bound is 168, but only.

We discuss the riemannhurwitz formula both in the case of riemann surfaces. An automorphism of a design is an isomorphism of a design with itself. The genus of compact riemann surfaces with maximal. Lecture 21 riemannhurwitz formula, chevalleys theorem. A theorem of hurwitz states that the maximal automorphism group of a compact riemann surface of genus 9 has order at most 84g1. Now suppose a finite group acts faithfully on a riemann surface. Finally, we state some bounds which also hold in positive. Hurwitz theorem 3 the basic example of a composition algebra over a. On the oikawa and arakawa theorems for graphs springerlink. Hurwitzs theorem gives a sharp bound on the size of automorphism groups, which is linear in the genus. In conclusion, we examine how fruchts theorem applies to the speci c case of cubic hamiltonian.

Under composition, the set of automorphisms of a graph forms what algbraists call a group. Bilinear transformations, their properties and classifications. See 1 for additional information about hurwitz curves. Propositions 1 and 3 below deal with certain properties of some or all hurwitz groups. The set of all automorphisms of a design form a group called the automorphism group of the design, usually denoted by autname of design. The supremum is readily seen to be nite by cauchys estimate, but we dont even. Pdf harmonic morphisms of graphs and the riemannhurwitz. We derive a closedform expression for all genus 1 hurwitz numbers, and give a simple new graphtheoretic interpretation of hurwitz numbers in genus 0 and 1. Automorphisms of riemann surfaces macsphere mcmaster. In certain cases these curves have automorphism group psl2. Riemannhurwitz formula, chevalleys theorem we begin with a remark on the tangent cone. Any automorphism of xthat is neither the identity nor the hyperelliptic involution must induce a nontrivial automorphism of p1 preserving s. The main aim of this article is to give an exposition of the diagrammatic proof due to boos and rost of the theorem of hurwitz that the dimension of a composition algebra is one of 0,1,2,4,8.

Recall that hurwitz groups, the automorphism groups of curves of genus g 2 with exactly 84g 1 automorphisms, are in onetoone correspondence with groups generated by two elements x. A bound for the number of automorphisms of an arithmetic riemann. By the riemannhurwitz formula if is a subgroup of index n in. Finally, an isomorphism has an inverse which is an isomorphism, so the inverse of an automorphism of gexists and is an automorphism of g. The automorphism group of a design is always a subgroup of the symmetric group on v letters where v is the number of points of the design. Recursions, formulas, and graphtheoretic interpretations ravi vakil abstract. Also, there are hurwitz curves with di erent automorphism groups besides those listed in theorem 1. Remark for example, f n 1 nz is holomorphic and has no zero in. It is wellknown that the klein quartic is the unique genus 3 curve that attains the hurwitz bound. Pdf nilpotent automorphism groups of riemann surfaces. For every prime number p 5 there exists a nal curve cwith autc.

If the f n are injective, and fis not constant, then fis injective. Symmetries of riemann surfaces on which psl2, q acts as. Note that f is dominant, hence, it is equivalent to give a. The celebrated hurwitz theorem 1893 states that the order of the. Hurwitz s original proof was stated for f c, but the eld of scalars only needs to be of characteristic not equal to 2 for his proof to work. Harmonic morphisms of graphs and the riemannhurwitz theorem article pdf available in doklady mathematics 931. Hurwitz s theorem claims that in fact more is true. Proof of hurwitzs automorphisms theorem mathoverflow. The inner product is then determined and is given by ha,bi ab.

A curve with the property that the size of its automorphism group attains the maximum bound for its genus is called a hurwitz curve. Proof of theorem 1 in each case, let sdenote the set of branch points of the separable 2to1 map x. Our goal is to prove this theorem using hurwitz quaternions. A riemannhurwitz theorem in graph theory oliver knill and tom tucker abstract. Applying the galois automorphism to f simply leads to applying the galois automorphism. In x5, we study the automorphism groups that are extremal for theorem 1. In number theory, hurwitzs theorem, named after adolf hurwitz, gives a bound on a diophantine approximation. G is a normal subgroup of autg the group of all automorphisms and the quotient is called the outer automorphism group of g. To complete the proof of the riemann mapping theorem, let ff ngbe a sequence from fsuch that lim n fjf0 n z 0jg sup f2f fjf0zjg.

Thus it corresponds to a finite separable morphism f. If gis a nite simple graph on which an arbitrary nite group a of order nacts by automorphisms. Generalization of a theorem of hurwitz 225 with cab algebraic. Preliminaries hurwitz series comorphisms automorphisms all rings are associative, commutative and unitary. The theorem states that for every irrational number. The main theorem of this paper is the following formula for. Hurwitzs automorphisms theorem bounds the order of the automorphism group of a negatively curved riemann in terms of the genus. The main object of investigation in the paper is automorphism groups of a graph. These include the theorems of hurwitz and rouche, the open mapping theorem, the inverse and implicit function theorems, applications of those theorems, behaviour at a critical point, analytic branches, constructing riemann surfaces for functional inverses, analytic continuation and monodromy, hyperbolic geometry and the riemann mapping theorem. Cornalba classi ed maximal strata, and we see that for g 3 every maximal. When we look at gas a rami ed cover over the chain h ga, the riemannhurwitz formula. Let be a smooth connected riemann surface of genus. Automorphisms of riemann surfaces mcmaster university.

In mathematics, hurwitzs automorphisms theorem bounds the order of the group of. Hurewicz theorem indicates that the hurewicz homomorphism induces an isomorphism between a quotient of the fundamental group and the rst homology group, which provides us with a lot of information about the fundamental group. Hurwitz states that the maximal automorphism group of a compact riemann. Then a g is perfect and therefore insoluble, and b g has a maximal normal subgroup k such that gk is non abelian simple hurwitz group. Hurwitz fs given by 1 is stable if and only if all the egg principal minors of the hurwitz matrix hn are positive and 30 0. Among the most interesting compact riemann surfaces are those with a group g of automorphisms which is relatively large compared with g. On a theorem of hurwitz cambridge university press. By the classical hurwitz theorem the group autx of holomorphic automorphism of x is a finite. En route, we interpret the genus 0 and 1 numbers as counting graphs with simple properties theorem 1.