On symmetric representations of groups of automorphism of bordered Klein surfaces

An automorphism of a bordered compact Klein surface induces a permutation of its boundary components and we study the corresponding representations of groups of automorphisms of such surfaces in the corresponding finite symmetric groups. The principal result gives a characterization of those abstract representations of a finite group G in symmetric groups which, up to the natural equivalence, proceed from compact Klein surfaces on which G acts as a group of dianalytic automorphisms. We deal with such representations for actions of G given by a so called smooth epimorphisms Λ → G, where Λ are so called non-Euclidean crystallographic groups (NEC-groups). For such data we calculate the kernels of our representations which allow, as an application, to get some results on the minimal degree of such representations for finite abstract groups.

such surfaces of given genus, and the groups themselves describe the nature of such singularities in some way. On the other hand there is a well known functorial equivalence between such surfaces and smooth, irreducible, complex, projective, algebraic curves and particular role play so called symmetric Riemann surfaces which correspond to complex curves having real forms. More precisely, under this equivalence, a Riemann surface X admits a symmetry σ , by which we understand an antiholomorphic involution, if and only if the corresponding curve C X has a real form C X (σ ). Furthermore, two such symmetries σ and τ define real forms C X (σ ) and C X (τ ), birationally isomorphic over the field R of real numbers, if and only if they are conjugate in the group Aut ± (X ) of all, including antiholomorphic, automorphisms of X and where Cent stands for the centralizer. Now for a Riemann surface X with a symmetry σ having nonempty set of fixed points, the orbit space X/σ is a bordered topological surface with, inherited from X , the natural dianalytic structure of so called Klein surface. Notice also, that each Klein surface arise from a symmetric Riemann surface in this way and a corresponding symmetry is unique up to conjugation in Aut ± (X ). A study of groups of dianalytic automorphisms of such surfaces is equivalent to a study of groups of birational automorphisms of real algebraic curves due to isomorphism Aut(X/σ ) ∼ = Aut R (C X (σ )) and the advantage of the former is made by a counterpart of the Riemann uniformization theorem, which together with some principal facts from covering theory allows their study through a well developed combinatorial theory of non-Euclidean crystallographic groups.
Next an automorphism of a bordered Klein surface induces a permutation of its boundary components and we study the associated representations of groups of automorphisms of such surfaces in the corresponding finite symmetric groups. The principal result gives a characterization of those abstract representations of a finite group G in symmetric groups which, up to the natural equivalence of symmetric representations, proceed from bordered Klein surfaces on which G acts as a group of dianalytic automorphisms. We deal with such representations for actions of finite groups G given by a so called smooth epimorphism → G, where are so called non-Euclidean crystallographic groups. For them, we calculate the kernels of the corresponding representations which allow, as an application, to get some illustrative examples and general results on the minimal degree of such representations as well. Our results describe in such a way, a nature of groups of automorphisms of real algebraic curves.

Preliminaries
We shall use a combinatorial approach based on non-Euclidean crystallographic groups (NEC-groups in short); we send the reader to the monographs [1] and [3] for detailed exposition of the whole theory.

Non-Euclidean crystallographic groups
An NEC-group is a discrete and cocompact subgroup of the group G of isometries of the hyperbolic plane H including those which reverse orientation. If such a subgroup contains only orientation preserving isometries, it is called a Fuchsian group.
(1) Particular role, in our considerations, will play the brackets C i = (n i1 , . . . , n is i ), called the period cycles with the numbers n i j ≥ 2, called the link periods. A group with signature (1) has the presentation with the following generators, called canonical generators : and relators: . . , s i and according to whether the sign is + or −. Generators c i, j are hyperbolic reflections and reflections c i, j−1 , c i, j corresponding to a link period n i, j are said to be consecutive or neighbouring.
Every NEC-group has associated a fundamental region, whose hyperbolic area μ( ) for an NEC-group with signature (1) is given by where ε = 2 if the sign is + and ε = 1 otherwise. Finally, it is known that an abstract group with the above presentation can be realized as an NEC-group with the signature (1) if and only if (2) is positive and if is a subgroup of finite index in an NEC-group then it is an NEC-group itself and there is a Hurwitz-Riemann formula, which says that

Riemann and Klein surfaces and their group of automorphisms
A Klein surface is a compact topological surface with a dianalytic structure which, in contrast to the classical analytic structure, involve the conjugation z →z for transition maps between charts and for the local forms of automorphisms between such surfaces. Similarly, as for Riemann surfaces, a bordered Klein surface of algebraic genus g ≥ 2 can be represented as the orbit space H/ with an NEC-group having a signature (g ; ±; . Furthermore a finite group G is a group of automorphisms of a surface so represented if and only if G ∼ = / for some NEC group and, generally, an epimorphism → G with the kernel being bordered surface NEC-group will be called smooth. Finally two dianalytic actions given by smooth epimorphisms θ : → G and θ : → G are topologically equivalent if and only if the diagram commutes for some isomorphisms ϕ : → and ψ : G → G .

Representations of groups of automorphisms of Klein surfaces on their boundary
Let G be a group of automorphisms of a compact Klein surface X with the boundary consisting of the connected components O 1 , . . . , O n , each of which turns out to be homeomorphic to a circle. We define ρ X : G → S n by This representation will be called geometric . With this definition we have the following The mapping ρ X is a homomorphism whose kernel is either cyclic or dihedral.
Proof If ρ X (g)(i) = i, then g induces an isometry of the circle O i . So the assertion follows from the fact that a finite group of isometries of a circle is either cyclic or dihedral, from [5,10] on the extension of automorphisms of bordered Klein surfaces and from the fact that an orientation preserving automorphism of a compact Riemann surface having a non-discrete set of fixed points is trivial.
As an immediate consequence we obtain the following result of Bujalance [2] about groups of automorphisms of bordered Klein surfaces with a small number of boundary components.

Corollary 3.2 The group of automorphisms of a Klein surface having 1 ≤ k ≤ 4 boundary components is solvable.
Here and throughout the remainder of the paper we shall use left hand side notation for conjugations and commutators i.e. a x = xax −1 and [a, x] = axa −1 x −1 .
Proof It is well known how to calculate the number of boundary components of a compact Klein surface X , in terms of θ and the signature of [3]. But here we shall need not only such quantitative results but also a qualitative description of boundary components. We shall use an approach developed in [6] and already successfully applied in [4] and [7] for example. The connected components of the boundary of X are in the bijective correspondence with the reflections of which are in and which are not conjugated in . On the other hand, each reflection of is conjugate to some canonical reflection from C. So we have to look for representatives of the conjugacy classes with respect to the conjugations by the elements of in the set which, in principle comes from Observe however that if c i ∼ c i , say c i = c λ i and c λ i , c λ i ∈ , then c λ i = c λ λ i . It follows that actually the canonical reflections in (4) can be runed over the set {c i : i ∈ I } from the statement. Now, given i ∈ I we have for some γ ∈ , which is equivalent to saying that λ −1 λ ∈ C( , c i ) , as is a normal subgroup of . Therefore conjugates of c i give rise to Thus h ∈ ker ρ X , which completes the proof.
Having calculated the kernel of the representation ρ X for the group G of dianalytic automorphisms of a bordered Klein surface X given by the smooth epimorphism θ : → G, we obtain at once the following, well known and obvious.

Corollary 3.4 Given a finite group G, there exists a bordered
Klein surface X such that G ⊆ Aut(X ) and ρ X is faithful.

Remark 3.5
There is a bijection between irreducible G-invariant subsets of the set of boundary components of X and the conjugacy classes in of canonical reflections of which are in . Furthermore, the subset corresponding to the class of a reflection c has [G : θ( C( , c))] elements. Indeed, (4) is the set of all reflections of . As we already observed, reflections c in (4) actually run over the set C of representatives of the conjugacy classes in of all canonical reflections of . Observe also that given c, c ∈ C which are in , their conjugations give boundary components in two disjoint G-invariant subsets. Finally, it is trivial that all conjugations of given c ∈ C are conjugate in and so the subset corresponding to c is irreducible, its cardinality is found in (5).

The characterization of geometric representations
We already have shown that given a finite group there is a bordered Klein surface X for which G is a group of dianalytic automorphisms and ρ X is faithful. It should be interesting to find for given G the minimum n = n(G) so that ρ(G) ⊂ S n . Here we shall prove our principal result which gives necessary and sufficient conditions for an abstract permutational representation ρ : G → S n to be equivalent to a representation ρ X proceeding from a bordered Klein surface X with G as a group of dianalytic automorphisms.
Remark 4. 1 We already know that the boundary components of X correspond to the conjugacy classes in of reflections in . Furthermore recall that given g = θ (λ) ∈ G and a boundary component O, say corresponding to the reflection c, we have where O is a boundary component corresponding to c λ .
be the non-conjugate in reflections corresponding to the elements O 1 , . . . , O n i of i . Then, given g = θ (λ) ∈ G, we have and therefore, since, by (5), θ(λ 1 ), . . . , θ (λ n i ) are representatives of all cosets in G/θ ( C ( , c i )), we see that the action of ρ X on i is regular with respect to θ( C( , c i )), which by [11,12], is either cyclic or dihedral. Conversely, assume that ρ = ρ 1 ⊕ . . . ⊕ ρ k is a decomposition of a representation ρ into irreducible representations ρ i of G in the symmetric group S m i on the sets i and suppose that ρ 1 , . . . , ρ s are regular with respect to dihedral subgroups Then for X = H/ , where = ker θ we have a representation ρ X : G → S n , where n = n 1 + · · · + n k , equivalent to ρ.

On the degree of geometrical representations of finite groups
Let G = / , where is an NEC group, is a surface NEC-group and let θ : → G be the corresponding smooth epimorphism. We know that given a finite group G there is a bordered Klein surface X with k boundary components for which ρ X : G → S k is faithful and a natural problem considered here is to find the minimal k in such situation. On the other hand having arbitrary elements a i and arbitrary involutions d i , d i for which (8) is minimal, let g 1 , . . . , g l be an arbitrary set of generators for G. Then an epimorphism constructed in the second part of Theorem 4.2 gives a Klein surface X with a group of automorphisms G for which ρ X is a geometrical permutation representation of G of the minimal degree.

Groups for which k(G) = |G|
It is obvious that k(G) ≤ |G| and here we shall look for the groups for which k(G) = |G|. We start with rather obvious Hence, there exists a geometrical representation of degree p α 1 1 + · · · + p α r r which is the minimum degree of a faithful permutation representation for a cyclic group of order N . If r = 1 then obviously the minimum degree of a geometrical representation is equal to N . The maximal subgroup of D N which has trivial intersection with G has order 2 and so it is generated by an involution. No element outside G is central, so g∈G 1≤i≤r Now suppose that N is a power of a prime, N > 2. Then for a noncentral involution d ∈ D N we have g∈G g −1 d g = 1.
Hence the minimum degree of a faithful geometrical representation is equal to | D N |/2 = N and is the same as the minimum degree of a faithful permutation representation.
It is well known that every nontrivial subgroup of Q 2 n contains a subgroup x 2 n−1 . Therefore we obtain the minimal value of (8) for r = 1 and a 1 = 1. Hence the minimum degree of a faithful geometrical representation is equal to 2 n+1 .

Proposition 5.5 If G is a finite group then k(G) = |G| if and only if G is one of the following groups:
(a) a cyclic group of order a power of a prime, (b) a generalized quaternion group, (c) an elementary abelian group of order 2 n .
Proof The implication ⇐ follows from the previous examples and we shall prove the converse. If |G| is divisible by two different primes p and q then it contains two cyclic subgroups of orders p and q respectively. Since they have trivial intersection, k(G) ≤ |G|/ p + |G|/q < |G|. Thus G is a p-group. If every two nontrivial subgroups of |G| have nontrivial intersection then either G is cyclic or p = 2 and G is a generalized quaternion group (see for instance [8], Satz 5.3.7). Suppose G has two subgroups a 1 and a 2 of order p m and p k respectively. Then k(G) ≤ |G| 1/ p m + 1/ p k . If the right hand side of this inequality is equal to |G|, then p = 2 and m = k = 1, which means that G is elementary abelian of order 2 n , for some n.
It is known that if we put n = 2 in the last item of the above proposition we get the description of groups G for which the minimum degree of a faithful permutation representation is equal to |G|.

On k(G) for finite simple groups at large
If G is a finite simple group then the sum from Theorem 5.1 has obviously only one component coming from a cyclic or dihedral subgroup of maximal order.

Lemma 5.6
An arbitrary maximal subgroup of a finite simple group is not cyclic.