Beauville Surfaces and Finite Simple Groups

نویسندگان

  • SHELLY GARION
  • Fritz Grunewald
  • ALEXANDER LUBOTZKY
چکیده

A Beauville surface is a rigid complex surface of the form (C1 × C2)/G, where C1 and C2 are non-singular, projective, higher genus curves, and G is a finite group acting freely on the product. Bauer, Catanese, and Grunewald conjectured that every finite simple group G, with the exception of A5, gives rise to such a surface. We prove that this is so for almost all finite simple groups (i.e., with at most finitely many exceptions). The proof makes use of the structure theory of finite simple groups, probability theory, and character estimates. Catanese [Cat] defined a Beauville surface to be an infinitesimally rigid complex surface of the form X := (C1 × C2)/G, where C1 and C2 are non-singular projective curves of genus ≥ 2, and G is a finite group acting freely on the product. Every g ∈ G respects the product decomposition C1 × C2. Let G denote the subgroup of G (of index ≤ 2) which preserves the ordered pair (C1, C2). Any Beauville surface can be presented in such a way that G acts effectively on each factor. Catanese called such a presentation minimal and proved that it is unique [Cat, Proposition 3.13]. A Beauville surface is mixed (resp. unmixed) if [G : G] = 2 (resp. G = G). There has been an effort to classify finite groups G appearing in minimal presentations of unmixed Beauville surfaces. A finite group G appears in this way if and only if it admits an unmixed Beauville structure. This consists of an ordered quadruple (x1, y1, x2, y2) ∈ G 1991 Mathematics Subject Classification. Primary 20D06; Secondary 14H30 14J10 20H10. SG was supported by a European Postdoctoral Fellowship (EPDI). ML was supported by grants from the National Science Foundation and the United States Israel Binational Science Foundation. AL was supported by grants from the European Research Council and the National Science Foundation. 1 2 SHELLY GARION, MICHAEL LARSEN, AND ALEXANDER LUBOTZKY such that each set {xi, yi} generates G and Σ(x1, y1) ∩ Σ(x2, y2) = {e}, where Σ(x, y) is the union of conjugacy classes of all powers of x, all powers of y, and all powers of xy. Bauer, Catanese, and Grunewald proved [BCG, Theorem 7.16] that all sufficiently large alternating groups admit an unmixed Beauville structure and conjectured [BCG, Conjecture 7.17] that all (non-abelian) finite simple groups except A5 do so. (The group A5 is a genuine exception.) Some progress has been made on this conjecture. In [FuGo] it was shown that it holds for alternating groups, i.e., that An admits an unmixed Beauville structure for all n ≥ 6. Partial results for groups of the form PSL2(q) and B2(2 ) were given in [BCG]; complete results for the same two families together with the Ree groups G2(3 ) were proved in [FuJo] and [GaPe]. Additionally, the groups of the formG2(q) and D4(q) (in characteristic p > 3) and PSL3(q) and PSU3(q) (all q) were treated in [GaPe]. The goal of the current paper is to prove the Bauer-Catanese-Grunewald conjecture for almost all finite simple groups. Theorem 1. For every sufficiently large non-abelian finite simple group G, there exists a Beauville surface with minimal presentation C1 × C2/G. Before saying something about the proof, let us put our result in a more general group-theoretic context. The existence of an unmixed Beauville structure on G amounts to the realization of G as a quotient of (hyperbolic) triangle groups in two “independent” ways. The question of which finite simple groups are quotients of triangle groups has attracted a lot of attention in the group theory literature. There has been special interest in the (2, 3, 7)-triangle group, also known as the Hurwitz group (see [Con] and the references therein), but considerable attention has been paid to other triangle groups as well (see [Mar] and the references therein). Usually the question has been asked from the angle of characterizing finite simple quotients of a fixed triangle group. Here the point of view is different: given a finite simple group, we will provide several (in fact, many) “triangle” generations of it. In this context, one should mention the old conjecture of Higman, proved by Everitt [Eve], asserting that for every hyperbolic triangle group Γ, all but finitely many alternating groups are quotients of Γ. Of course, this gives Theorem 1 for almost all the alternating groups. One should not be tempted to conjecture that given Γ, almost all finite simple groups BEAUVILLE SURFACES AND FINITE SIMPLE GROUPS 3 are quotients of Γ. In this connection, [Mar] has some intriguing and suggestive results. Taking an even broader perspective, one can study finite quotients and representations of Fuchsian groups (see, e.g., [LiSh1] and [LaLu]), but usually triangle groups are the most difficult to deal with. Let us now give a short description of the proof. As mentioned above, the result is known for alternating groups, and we can ignore the finitely many sporadic groups. By the classification of finite simple groups, we need only consider groups of Lie type. For each such group which is sufficiently large, we will present two maximal tori T1 and T2. If Ci denotes the set of all conjugates of elements of Ti, we ensure that C1 ∩ C2 = {e}, and that in each Ci we can find xi and yi such that (a) G = 〈xi, yi〉, (b) xiyi ∈ Ci. To assure (a), we use some of the well developed theory bounding the indices and the number of maximal subgroups in G. On the other hand, (b) can be proved by means of character estimates. While it is quite likely that the estimates we need can be deduced from Lusztig’s theory, a much softer and more elementary method is presented in §1. This method works for general finite groups and may have some independent interest and applications. The paper is organized as follows. In §1, the new character estimate is presented. In §2, we state Proposition 7, which claims that two tori exist with some desired properties. We illustrate the proposition by proving it for groups of the form PSLn(q) for all but finitely many pairs (n, q). We then present Theorem 1 modulo Proposition 7, leaving the detailed case analysis required to prove Proposition 7 in general to §3. The only groups not covered by prior work are now the Ree groups F4(2 ), which are discussed in §4, using a different method which is applicable to groups whose maximal subgroup structure is well understood. This paper is dedicated to the memory of Fritz Grunewald, who has been influential in the area of Beauville surfaces by connecting it to group theory [BCG]. It has been typical of Fritz to serve as a bridge between different areas of mathematics. His legacy as a mathematician and as an outstanding personality will be remembered for many years. Acknowledgment. The first-named author would like to thank Ingrid Bauer, Fabrizio Catanese and Fritz Grunewald for introducing her to the fascinating world of Beauville structures and for many useful discussions. She is also grateful to Bob Guralnick and Eugene Plotkin for many helpful discussions. 4 SHELLY GARION, MICHAEL LARSEN, AND ALEXANDER LUBOTZKY 1. Elementary Character Bounds for General Finite Groups Let G be any finite group. We say that a ∈ G is abstractly regular if the centralizer Z(a) of a in G is abelian. We give an upper bound for the absolute value of irreducible characters evaluated at abstractly regular elements. Lemma 2. Let A denote a maximal abelian subgroup of G and N = N(A) its normalizer in G. If a ∈ A is abstractly regular, then gag ∈ A if and only if g ∈ N . Proof. As A is abelian and a ∈ A, A ⊂ Z(a). As a is abstractly regular, Z(a) is abelian. Since A is maximal abelian, A = Z(a). Thus, gAg = gZ(a)g = Z(gag). If gag ∈ A, then Z(gag) contains A. Since |Z(gag)| = |Z(a)| = |A|, we have gAg = gZ(a)g = Z(gag) = A, so g ∈ N . The converse is trivial. Theorem 3. Let A denote a maximal abelian subgroup of G. Suppose that there exist proper subgroups A1, . . . , An of A such that every element in A := A \ (A1 ∪ A2 ∪ · · · ∪ An) is abstractly regular. Let N = N(A) denote the normalizer of A. If χ is any irreducible character of G, then |χ(a)| ≤ (4/ √ 3)[N : A] for all a ∈ A. Proof. Let A = Hom(A,C). We identify ZA with the ring of characters of virtual complex representations on A. Let φ ∈ ZA be the character associated to the restriction of χ to A. For each i from 1 to n, let φi ∈ A be a character which is trivial on Ai and non-trivial on a. For every non-trivial root of unity ω, there exists an integer k such that |ωk − 1| ≥ √ 3, so replacing φi by a character of the form φ k i , we may assume that |φi(a)− 1| ≥ √ 3. Let

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

New Beauville Surfaces and Finite Simple Groups

In this paper we construct new Beauville surfaces with group either PSL(2, p), or belonging to some other families of finite simple groups of Lie type of low Lie rank, or an alternating group, or a symmetric group, proving a conjecture of Bauer, Catanese and Grunewald. The proofs rely on probabilistic group theoretical results of Liebeck and Shalev, on classical results of Macbeath and on recen...

متن کامل

Beauville Surfaces and Probabilistic Group Theory

A Beauville surface is a complex algebraic surface that can be presented as a quotient of a product of two curves by a suitable action of a finite group. Bauer, Catanese and Grunewald have been able to intrinsically characterize the groups appearing in minimal presentations of Beauville surfaces in terms of the existence of a so-called ”Beauville structure“. They conjectured that all finite sim...

متن کامل

Beauville p-groups: a survey

Beauville surfaces are a class of complex surfaces defined by letting a finite group G act on a product of Riemann surfaces. These surfaces possess many attractive geometric properties several of which are dictated by properties of the group G. In this survey we discuss the p-groups that may be used in this way. En route we discuss several open problems, questions and conjectures.

متن کامل

Beauville Surfaces, Moduli Spaces and Finite Groups

In this paper we give the asymptotic growth of the number of connected components of the moduli space of surfaces of general type corresponding to certain families of Beauville surfaces with group either PSL(2, p), or an alternating group, or a symmetric group or an abelian group. We moreover extend these results to regular surfaces isogenous to a higher product of curves.

متن کامل

More on Strongly Real Beauville Groups

Beauville surfaces are a class of complex surfaces defined by letting a finite group G act on a product of Riemann surfaces. These surfaces possess many attractive geometric properties several of which are dictated by properties of the group G. A particularly interesting subclass are the ‘strongly real’ Beauville surfaces that have an analogue of complex conjugation defined on them. In this sur...

متن کامل

ذخیره در منابع من


  با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

عنوان ژورنال:

دوره   شماره 

صفحات  -

تاریخ انتشار 2010