ON BEAUVILLE STRUCTURES FOR PSL(2, q)

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1 ON BEAUVILLE STRUCTURES FOR PSL(, q) SHELLY GARION Abstract. We characterize Beauville surfaces of unmixe type with group either PSL(, p e ) or PGL(, p e ), thus extening previous results of Bauer, Catanese an Grunewal, Fuertes an Jones, an Penegini an the author. 1. Introuction 1.1. Beauville structures. A Beauville surface S (over C) is a particular kin of surface isogenous to a higher prouct of curves, i.e., S = (C 1 C )/G is a quotient of a prouct of two smooth curves C 1 an C of genera at least two, moulo a free action of a finite group G, which acts faithfully on each curve. For Beauville surfaces the quotients C i /G are isomorphic to P 1 an both projections C i C i /G = P 1 are coverings branche over three points. A Beauville surface is in particular a minimal surface of general type. Beauville surfaces were introuce by F. Catanese in [4], inspire by a construction of A. Beauville (see [3]). We have two cases: the mixe case where the action of G exchanges the two factors (an then C 1 an C are isomorphic), an the unmixe case where G acts iagonally on their prouct. In the following we shall consier only the unmixe case. Working out the efinition of an unmixe Beauville surface one sees that there is a purely group theoretical criterion which characterizes the groups of unmixe Beauville surfaces: the existence of what in [] is calle an unmixe Beauville structure. Definition 1.1. An unmixe Beauville structure for a finite group G consists of two triples (a 1, b 1, c 1 ) an (a, b, c ) of elements in G which satisfy (i) a 1 b 1 c 1 = 1 an a b c = 1, (ii) a 1, b 1 = G an a, b = G, (iii) Σ(a 1, b 1, c 1 ) Σ(a, b, c ) = {1}, where Σ(a i, b i, c i ) := {ga j i g 1, gb j i g 1, gc j i g 1 } for i = 1,. g G j=1 Moreover, τ i := (or(a i ), or(b i ), or(c i )) is calle the type of (a i, b i, c i ) (for 1 i = 1, ), an a type which satisfies the conition or(a i ) + 1 or(b i ) + 1 or(c i ) < 1 is calle hyperbolic. In this case, we say that G amits an unmixe Beauville structure of type (τ 1, τ ). 000 Mathematics Subject Classification. 0D06,0H10,14J9,30F99. The author was supporte by a European Postoctoral Fellowship (EPDI). 1

2 SHELLY GARION It is known that the following finite almost simple groups amit an unmixe Beauville structure: (a) The symmetric group S n, if an only if n 5 []; (b) The alternating group A n, if an only if n 6 [, 10]; (c) PSL(, q), where q = p e is a prime power, if an only if q 7 [, 9, 11]; () Suzuki groups Sz(q), where q = e+1, an Ree groups R(q), where q = 3 e+1 [11]; (e) Some other finite simple groups G(q) of Lie type of low Lie rank, such as PSL(3, q) an PSU(3, q), provie that q = p e is large enough [9]. Moreover, in [9] it is prove that if (r 1, s 1, t 1 ) an (r, s, t ) are two hyperbolic types, then almost all alternating groups A n amit an unmixe Beauville structure of type ( (r 1, s 1, t 1 ), (r, s, t ) ). This was previously conjecture by Bauer, Catanese an Grunewal in []. Analogously for PSL(, q), where q = p e is a prime power, the aim of this paper is to generalize the results given in [9, 11], an characterize the possible types of an unmixe Beauville structure for the group PSL(, q). The question of which finite groups amit an unmixe Beauville structure is eeply relate to the question of which finite groups are quotients of certain triangle groups. Inee, conitions (i) an (ii) of Definition 1.1 are equivalent to the conition that G is a quotient of each of the triangle groups (or(a i ), or(b i ), or(c i )) for i = 1, with torsion-free kernel. Therefore, we shall now recall some results regaring finite quotients of triangle groups. Note that it is conition (iii) of Definition 1.1 which makes the existence of an unmixe Beauville structure for G a more elicate issue. 1.. Triangle groups an their finite quotients. Starting with three positive integers r, s, t, consier the group (r, s, t) presente by the generators an relations (r, s, t) = x, y : x r = y s = (xy) t = 1, known as a triangle group. The triple (r, s, t) is hyperbolic if 1 r + 1 s + 1 t < 1. In this case, the corresponing triangle group (r, s, t) is also calle hyperbolic. Hyperbolic triangle groups are infinite. It is therefore interesting to stuy their finite quotients, particularly the simple ones therein. Among all hyperbolic triples (r, s, t), the triple (, 3, 7) attains the smallest positive value of 1 ( 1 r + 1 s + 1 ) t. Therefore, the stuy of the group (, 3, 7), known as the Hurwitz triangle group, an its finite quotients, known as Hurwitz groups, has attracte much attention, see for example [6] for a historical survey, an [1] an the references therein for the current state of the art. It was shown by Coner [5] (following Higman) that the alternating group A n is a Hurwitz group if n 168. Concerning the group PSL(, p e ), Macbeath [15] has shown that it is a Hurwitz group if an only if either e = 1 an p 0, ±1 (mo 7), or e = 3 an p ±, ±3 (mo 7). We thus see ifferent behaviors for the ifferent families of simple groups. Namely, any large enough alternating group is a Hurwitz group, whereas for

3 ON BEAUVILLE STRUCTURES FOR PSL(, q) 3 PSL(, p e ), the prime p etermines a unique exponent e such that PSL(, p e ) is a Hurwitz group. More generally, Higman ha alreay conjecture in the late 1960s that every hyperbolic triangle group has all but finitely many alternating groups as quotients. This was eventually prove by Everitt [8]. Later, Liebeck an Shalev [14] gave an alternative proof base on probabilistic group theory. Langer an Rosenberger [1] an Levin an Rosenberger [13] ha generalize the above result of Macbeath, an etermine, for a given prime power q = p e, all the triples (r, s, t) such that PSL(, q) is a quotient of (r, s, t), with torsion-free kernel. It follows that if (r, s, t) is hyperbolic, then for almost all primes p, there is precisely one group of the form PSL(, p e ) or PGL(, p e ) which is a homomorphic image of (r, s, t) with torsion-free kernel. This result will be escribe in etail is Section.1. We note that it can also be obtaine by using other techniques. Firstly, Marion [16] has recently provie a proof for the case where r, s, t are primes relying on probabilistic group theoretical methos. Seconly, it also follows from the representation theoretic arguments of Vincent an Zalesski [3]. Such methos can be use for ealing with other families of finite simple groups of Lie type, see for example [17, 18, 0, 3] Organization. This paper is organize as follows. Section presents the main Theorems in etail. In Section 3 we present some of the basic properties of the groups PSL(, q) an PGL(, q) that are neee later for the proofs. The proofs themselves are presente in Section 4. Acknowlegement. I woul like to thank Ingri Bauer, Fabrizio Catanese an Fritz Grunewal for introucing me to the fascinating worl of Beauville structures an for many useful iscussions. I am very thankful to Alexanre Zalesski for his interesting comments. I am grateful to Claue Marion for kinly proviing me with his recent preprints, an to Matteo Penegini for many interesting iscussions an for his remarks on this manuscript.. Main Theorems.1. Which triangle groups surject onto PSL(, q)? Notation.1. For a prime p an n N such that gc(n, p) = 1, efine (i) µ PGL (p, n) = min { f > 0 : p f ±1 (mo n) } { ; min { f > 0 : p f ±1 (mo n) } if n is o (ii) µ PSL (p, n) = min { f > 0 : p f ±1 (mo n) } if n is even ; (iii) We also set µ PGL (p, p) = 1 an µ PSL (p, p) = 1. Note that µ PGL (p, n) (respectively µ PSL (p, n)) is equal to the minimal integer e such that PGL(, p e ) (respectively PSL(, p e )) contains an element of orer n. Notation.. For a prime p an integers n 1,..., n k, such that each of them is either relatively prime to p or equal to p, efine (i) µ PGL (p; n 1,..., n k ) = lcm ( µ PGL (p, n 1 ),..., µ PGL (p, n k ) ). (ii) µ PSL (p; n 1,..., n k ) = lcm ( µ PSL (p, n 1 ),..., µ PSL (p, n k ) ).

4 4 SHELLY GARION Note that µ PGL (p; n 1,..., n k ) (respectively µ PSL (p; n 1,..., n k )) is equal to the minimal integer e such that PGL(, p e ) (respectively PSL(, p e )) contains k elements of orers n 1,..., n k. When q is o, one nees to istinguish triples (r, s, t) of orers of elements that generate PSL(, q) from the ones that generate PGL(, q). The latter triples are calle irregular (see [15, 9] an [1, Lemma 3.5]), an contain exactly two orers of elements in PGL(, q) \ PSL(, q) an one orer of an element in PSL(, q). More precisely, they are efine as follows. Definition.3. Let p be an o prime, an let (r, s, t) be a hyperbolic triple such that each of r, s, t is either relatively prime to p or equal to p. We say that (r, s, t) is irregular if there is a permutation (r, s, t ) of (r, s, t) such that one of the following cases occurs. Case (α): r, s, t >, r, s an e = µ PSL (p; r, s, t ) are all even, both µ PGL (p, r ) an µ PGL (p, s ) ivie e, both µ PSL (p, r ) an µ PSL (p, s ) o not ivie e, µ PSL (p, t ) ivies e. Case (β): r, s > an t =, r, s an e = µ PSL (p; r, s ) are all even, both µ PGL (p, r ) an µ PGL (p, s ) ivie e, both µ PSL (p, r ) an µ PSL (p, s ) o not ivie e. Case (γ): r, s >, an t =, r an e = µ PSL (p; r, s ) are even, µ PGL (p, r ) ivies e, µ PSL (p, r ) oes not ivie e, µ PSL (p, s ) ivies e. The following theorems summarize the results in [1, Theorems 4.1 an 4.] an [13, Theorems 1 an ]. Theorem A. [1, 13]. Let p be a prime an assume that q = p e is at least 7. Let r, s, t N. Then PSL(, q) is a quotient of (r, s, t) with torsion-free kernel if an only if (r, s, t) is hyperbolic an satisfies one of the conitions in the following table: p (r, s, t) e further conitions p 5 (p, p, p) 1 - p 3 permutation of (p, p, t ) µ PSL (p, t ) - gc(t, p) = 1 p 3 permutation of (p, s, t ) µ PSL (p; s, t ) either at most one of r, s, t is even, gc(s t, p) = 1 or: p 3 gc(r s t, p) = 1 µ PSL (p; r, s, t) if at least two of r, s, t are even, then none of (α), (β), (γ) occurs p = - µ PSL (; r, s, t) -

5 ON BEAUVILLE STRUCTURES FOR PSL(, q) 5 Theorem B. [1, 13]. Let p be an o prime an assume that q = p e is at least 5. Let r, s, t N. Then PGL(, q) is a quotient of (r, s, t) with torsion-free kernel if an only if (r, s, t) is hyperbolic an satisfies one of the conitions in the following table: (r, s, t) e further conitions permutation of (p, s, t ) gc(s t, p) = 1 gc(r s t, p) = 1 µ PSL (p;s,t ) at least two of r, s, t are even, an µ PSL (p;r,s,t) one of (α), (β), (γ) occurs The following corollary follows immeiately from Theorems A an B. Corollary C. Let p be a prime an let (r, s, t) be a hyperbolic triple such that each of r, s, t is either relatively prime to p or equal to p. Then there exist a unique exponent e an a unique G {PSL, PGL} such that G(, p e ) is a quotient of (r, s, t) with torsion-free kernel, namely (a) PSL(, p e ) where e = µ PSL (p; r, s, t), if (r, s, t) satisfies the conitions of Theorem A. (b) PGL(, p e ) where e = µ PSL(p;r,s,t), if (r, s, t) satisfies the conitions of Theorem B. Remark.4. For completeness, we list below the results for PSL(, q) where q < 7 an for PGL(, q) where q < 5. For each group in the table below, we list all the triples r s t such that (r, s, t) maps onto it with torsion-free kernel (see also [15, 8]). group triple(s) PSL(, ) = S 3 (,, 3) PSL(, 3) = A 4 (, 3, 3), (3, 3, 3) PGL(, 3) = S 4 (, 3, 4), (3, 4, 4) PSL(, 4) = PSL(, 5) = A 5 (, 3, 5), (, 5, 5), (3, 3, 5), (3, 5, 5), (5, 5, 5).. Beauville Structures for PSL(, q) an PGL(, q). We present here our result concerning the possible types of unmixe Beauville structures for PSL(, q). Theorem D. Let p be a prime an assume that q = p e is at least 7. Let (r 1, s 1, t 1 ) an (r, s, t ) be two triples of integers. Then the following conitions are sufficient to guarantee that the group PSL(, q) amits an unmixe Beauville structure of type ( (r 1, s 1, t 1 ), (r, s, t ) ) : (i) (r 1, s 1, t 1 ) an (r, s, t ) are hyperbolic. (ii) Each of (r 1, s 1, t 1 ) an (r, s, t ) satisfy one of the conitions of Theorem A. (iii) r 1 s 1 t 1 is relatively prime to r s t. These conitions are also necessary if either p = or p is o an e is o. When p is o an e is even, conitions (i),(ii) together with the following conition (iii ) are necessary. (iii ) gc(r 1 s 1 t 1, r s t ) ivies p.

6 6 SHELLY GARION Note that Beauville structures for PSL(, p e ) of type ( (p, p, t 1 ), (p, p, t ) ) o occur (when p is o an e is even), hence conition (iii ) cannot be improve (see Lemma 4.5). Our next result characterizes the possible unmixe Beauville structures for PGL(, q). Theorem E. Let p be a prime an assume that q = p e is at least 5. Let (r 1, s 1, t 1 ) an (r, s, t ) be two triples of integers. Then the group PGL(, q) amits an unmixe Beauville structure of type ( (r 1, s 1, t 1 ), (r, s, t ) ) if an only if the following conitions hol: (i) (r 1, s 1, t 1 ) an (r, s, t ) are hyperbolic. (ii) Each of (r 1, s 1, t 1 ) an (r, s, t ) satisfy one of the conitions of Theorem B. (iii) Each of the numbers gc(r 1, r ), gc(r 1, s ), gc(r 1, t ), gc(s 1, r ), gc(s 1, s ), gc(s 1, t ), gc(t 1, r ), gc(t 1, s ), gc(t 1, t ) equals 1 or. (iv) All even elements in one of the triples ivie q 1, while all even elements in the other triple ivie q + 1. (v) If one of the triples contains an element t =, then this triple must contain an even element r > an a thir element s >, an moreover: (a) If q 1 (mo 4) an r ivies q 1, then Case (β) hols; (b) If q 1 (mo 4) an r ivies q + 1, then Case (γ) hols; (c) If q 3 (mo 4) an r ivies q 1, then Case (γ) hols; () If q 3 (mo 4) an r ivies q + 1, then Case (β) hols. 3. Preliminaries In this section we shall escribe some well-known properties of the groups PSL(, q) an PGL(, q) (see for example [19, 6]) Definition. Let K be a fiel. Recall that GL(, K) is the group of invertible matrices over K, an SL(, K) is the subgroup of GL(, K) comprising the matrices with eterminant 1. Then PGL(, K) an PSL(, K) are the quotients of GL(, K) an SL(, K) by their respective centers. Let q = p e, where p is a prime an e 1. We enote the finite fiel of size q by F q. The algebraic closure of F p (which is equal to the algebraic closure of F q ) will be enote by F p. For simplicity, we shall enote by GL(, q), SL(, q), PGL(, q) an PSL(, q) the groups GL(, F q ), SL(, F q ), PGL(, F q ) an PSL(, F q ), respectively. When q is even, then one can ientify PSL(, q) with SL(, q) an also with PGL(, q), an so its orer is q(q 1)(q + 1). When q is o, the orers of PGL(, q) an PSL(, q) are q(q 1)(q + 1) an 1 q(q 1)(q + 1) respectively, an therefore we can ientify PSL(, q) with a normal subgroup of inex in PGL(, q). Moreover, PSL(, q) an PGL(, q) can be viewe as subgroups of PSL(, F p ). Recall that PSL(, q) is simple for q, 3.

7 ON BEAUVILLE STRUCTURES FOR PSL(, q) Group elements. One can classify the elements of PSL(, q) accoring to the possible Joran forms of their pre-images in SL(, q). The following table lists the three types of elements, accoring to whether the characteristic polynomial P (λ) := λ αλ + 1 of the matrix A SL(, q) (where α is the trace of A) has 0, 1 or istinct roots in F q. element roots canonical form in orer conjugacy classes type of P (λ) SL(, F p ) ( ) two conjugacy classes ±1 1 unipotent 1 root p in PSL(, q), which 0 ±1 ( α = ± ) unite in PGL(, q) a 0 split roots 0 a 1 ivies 1 (q 1) for each α: where a F q = 1 for q even one conjugacy class an ( a + a 1 ) = α = for q o in PSL(, q) a 0 non-split no roots 0 a q ivies 1 (q + 1) for each α: where a F q \ F q = 1 for q even one conjugacy class a q+1 = 1 = for q o in PSL(, q) an a + a q = α Recall that if p is o an q = p e, then any element in PGL(, q) is either of orer p ( unipotent ) or of orer iviing q 1 ( split ), or of orer iviing q + 1 ( non-split ). Moreover, any element which belongs to PGL(, q) but not to PSL(, q) has an even orer iviing either q 1 but not q 1 or q + 1 but not q Elements of orer. Note that all( elements ) of orer in PSL(, q) 0 1 are conjugate to the image of the matrix. They are unipotent if 1 0 p =, split if q 1 (mo 4), an non-split if p 3 (mo 4). Moreover, if q is an o prime power, then PGL(, q) always contains elements of orer which are not containe in PSL(, q). These elements are split if q 3 (mo 4), an non-split if q 1 (mo 4). Therefore, if q is o, then an element of orer in a hyperbolic irregular triple satisfies exactly one of the following: q 1 (mo 4) q 3 (mo 4) Case (β) is split is non-split Case (γ) is non-split is split 4. Beauville Structures for PSL(, q) an PGL(, q) In this Section we prove Theorems D an E.

8 8 SHELLY GARION 4.1. Cyclic groups. The following easy Lemma is neee for the proof of Theorems D an E. Lemma 4.1. Let C be a finite cyclic group, an let x an y be non-trivial elements in C. If the orers of x an y are not relatively prime, then there exist some integers k an l such that x k = y l 1. Proof. Denote the orers of x an y by a an b respectively, then, by assumption, gc(a, b) = 1, an so one can write a = a an b = b, where gc(a, b ) = 1. Hence, x a an y b are of exact orer. Observe that C has only one cyclic subgroup of orer, an let z be a generator of this subgroup. Thus, x a = z = y b. Therefore, there exist some integers k an l such that x a k = z = y b l. 4.. Elements an conjugacy classes in PSL(, q) an PGL(, q). Notation 4.. For a finite group G an a 1,..., a n G, efine Σ(a 1,..., a n ) = {ga j 1 g 1,..., ga j ng 1 }. g G j=1 Note that for n = 3 this notation coincies with the one given in Definition 1.1(iii). Observe that for a 1,..., a n, b 1,..., b m the conition is equivalent to the conition that Σ(a 1,..., a n ) Σ(b 1,..., b m ) = {1} Σ(a i ) Σ(b j ) = {1} for every 1 i n, 1 j m. Lemma 4.3. Let q = p e be a prime power an let A 1, A PSL(, q). Then Σ(A 1 ) Σ(A ) = {1} if an only if one of the following occurs: (1) The orers of A 1 an A are relatively prime; () p is o, e is even, or(a 1 ) = p = or(a ) an A 1, A are not conjugate in PSL(, q). Proof. If the orers of A 1 an A are relatively prime then every two nontrivial powers A i 1 an Aj have ifferent orers, thus {g 1 A i 1g1 1 } g 1,i {g A j g 1 } g,j = {1}, as neee. Now, assume that the orers of A 1 an A are not relatively prime. If there exists some prime r p which ivies the orers of A 1 an A, then r ivies exactly one of q 1 or q+1, where = 1 if p = an = if p is o, since q 1 an q+1 are relatively prime. Hence, the orers of A 1 an A both ivie exactly one of q 1 or q+1, an so A 1 an A can be conjugate in PSL(, q) to two elements which belong to the same cyclic group (either of orer q 1 or of orer q+1 ). Lemma 4.1 now implies that

9 ON BEAUVILLE STRUCTURES FOR PSL(, q) 9 there exist some integers i an j such that A i 1 an Aj are conjugate in PSL(, q), an so Σ(A 1 ) Σ(A ) {1}. If or(a 1 ) = p = or(a ) then A 1 an A are unipotents, an so, for k = 1, (, A k ) can be conjugate in PSL(, q) to the image of some matrix 1 A k = ak, where a 0 1 1, a F q. Recall that if q = e then A 1 an A are always conjugate in PSL(, q), an if q = p e is o, then A 1 an A are conjugate in PSL(, q) if an only if either both a 1 an a are squares in F q or both of them are non-squares. Note that if p is o an e is even then all the elements {k : 1 k p 1} are squares in F q. If p is o an e is o, then half of the elements {k : 1 k p 1} are squares in F q an half are non-squares. Therefore, if q = e, then A 1 an A are necessarily conjugate, an so Σ(A 1 ) Σ(A ) {1}. If ( p is o ) an e is even, then for any 1 i, j p 1, A i 1 is( conjugate ) 1 ia1 1 to, which is conjugate to A 0 1 1, an Aj is conjugate to ja, 0 1 which is conjugate to A. Hence, Σ(A 1) Σ(A ) {1} if an only if either both a 1 an a are squares in F q or both of them are non-squares, namely, if an only if A 1 an A are conjugate. If p is o an e is o, we can choose 1 i, j p 1 as follows: i = 1 if a 1 is a square, an i is a non-square in F q otherwise, j = 1 if a is a square, an j is a non-square in F q otherwise, ( ) 1 1 an so, both A i 1 an Aj are conjugate in PSL(, q) to the image of, 0 1 implying that Σ(A 1 ) Σ(A ) {1}. Lemma 4.4. Let q = p e be an o prime power an let A 1, A PGL(, q). Then Σ(A 1 ) Σ(A ) = {1} if an only if one of the following occurs: (1) gc(or(a 1 ), or(a )) = 1; () A 1 is split, A is non-split an gc(or(a 1 ), or(a )) = ; (3) A 1 is non-split, A is split an gc(or(a 1 ), or(a )) =. Proof. If gc(or(a 1 ), or(a )) = 1 then every two non-trivial powers A i 1 an A j have ifferent orers, thus Σ(A 1) Σ(A ) = {1}, as neee. If A 1 is split an A is non-split, then necessarily gc(or(a 1 ), or(a )), since gc(q 1, q + 1) =. In this case, any non-trivial power of A 1 is a split element, while any non-trivial power of A is a non-split element, an so they are not conjugate in PGL(, q), implying that Σ(A 1 ) Σ(A ) = {1} as neee. If gc(or(a 1 ), or(a )) = an both A 1 an A are split (resp. nonsplit), then A 1 an A can be conjugate in PGL(, q) to two elements which belong to the same cyclic group of orer q 1 (resp. q + 1). Lemma 4.1 now implies that there exist some integers i an j such that A i 1 an Aj are conjugate in PGL(, q), an so Σ(A 1 ) Σ(A ) {1}.

10 10 SHELLY GARION If or(a 1 ) = p = or(a ), then A 1 an A are unipotents, ( an ) so they 1 1 can be conjugate in PGL(, q) to the image of the matrix, implying 0 1 that Σ(A 1 ) Σ(A ) {1}. Otherwise, gc(or(a 1 ), or(a )) = r, where r > an (r, p) = 1, an so r ivies exactly one of q 1 or q + 1, since gc(q 1, q + 1) =, implying that or(a 1 ) an or(a ) both ivie exactly one of q 1 or q + 1. Hence, A 1 an A can be conjugate in PGL(, q) to two elements which belong to the same cyclic group (either of orer q 1 or of orer q + 1). Lemma 4.1 now implies that there exist some integers i an j such that A i 1 an Aj are conjugate in PGL(, q), an so Σ(A 1 ) Σ(A ) {1} Proof of Theorem D. The conitions are sufficient. Let (r 1, s 1, t 1 ) an (r, s, t ) be two triples of integers. Assume that PSL(, q) is a quotient of the triangle groups (r 1, s 1, t 1 ) an (r, s, t ) with torsion-free kernel. Then one can fin elements A 1, B 1, C 1, A, B, C PSL(, q) of orers r 1, s 1, t 1, r, s, t respectively, such that A 1 B 1 C 1 = I = A B C an A 1, B 1 = PSL(, q) = A, B, an so conitions (i) an (ii) of Definition 1.1 are fulfille. Moreover, the conition that r 1 s 1 t 1 is relatively prime to r s t implies that each of r 1, s 1, t 1 is relatively prime to each of r, s, t, an so by Lemma 4.3, Σ(A 1, B 1, C 1 ) Σ(A, B, C ) = {1}, hence conition (iii) of Definition 1.1 is fulfille. Therefore, PSL(, q) amits an unmixe Beauville structure of type ( (r 1, s 1, t 1 ), (r, t, s ) ). The conitions are necessary. Assume that the group PSL(, q) amits an unmixe Beauville structure of type ( (r 1, s 1, t 1 ), (r, t, s ) ). Then there exist A 1, B 1, C 1, A, B, C PSL(, q) of orers r 1, s 1, t 1, r, s, t respectively, such that A 1 B 1 C 1 = I = A B C an A 1, B 1 = PSL(, q) = A, B, implying that PSL(, q) is a quotient of the triangle groups (r 1, s 1, t 1 ) an (r, s, t ) with torsion-free kernel, an so conitions (i) an (ii) are necessary. Moreover, Σ(A 1, B 1, C 1 ) Σ(A, B, C ) = {1}, an so by Lemma 4.3, if either p = or p is o an e is o, then each of r 1, s 1, t 1 is necessarily relatively prime to each of r, s, t, implying that r 1 s 1 t 1 is relatively prime to r s t. If p is o an e is even then, by Lemma 4.3, gc(r 1, r ) = 1 or p, gc(r 1, s ) = 1 or p, gc(r 1, t ) = 1 or p, gc(s 1, r ) = 1 or p, gc(s 1, s ) = 1 or p, gc(s 1, t ) = 1 or p, gc(t 1, r ) = 1 or p, gc(t 1, s ) = 1 or p, an gc(t 1, t ) = 1 or p. Moreover, it is not possible that (r 1, s 1, t 1 ) = (p, p, p) = (r, s, t ), since in this case e = 1 (by Theorem A). Thus, gc(r 1 s 1 t 1, r s t ) ivies p. The following Lemma shows that in case p o an e even, the conition that gc(r 1 s 1 t 1, r s t ) ivies p cannot be improve. Lemma 4.5. Let p be an o prime an let q = p e. Then the group PSL(, q ) amits an unmixe Beauville structure of type ( (p, p, t 1 ), (p, p, t ) ) where t 1 q 1 an t q +1.

11 Proof. Observe that the set ON BEAUVILLE STRUCTURES FOR PSL(, q) 11 D := {a 4 : a F q, a F q \ F q } contains both squares an non-squares in F q. Hence, there exist b, c F q such that b, c F q \ F q, c 4 is a square an b 4 is a non-square. Let x be a generator of the multiplicative group F q an let = b/x. Define the following matrices ( ) ( ) x A 1 =, A 0 1 =, 0 1 ( ) ( ) g 1 =, g c 1 =, 1 ( ) ( ) B 1 = ga 1 g 1 c = c, B c + 1 = ga 1 g 1 x + 1 x =, x x + 1 ( ) ( C 1 = (A 1 B 1 ) 1 c + 1 c = c c, C c + 1 = (A B ) 1 x + 1 x = ) x x x. x + 1 Now, one nees to verify that ( (Ā1, B 1, C 1 ), (Ā, B, C ) ), where Ā1, B 1, C 1, Ā, B, C are the images of A 1, B 1, C 1, A, B, C in PSL(, q ), is an unmixe Beauville structure for PSL(, q ). (i) A 1 B 1 C 1 = 1 = A B C an so Ā1 B 1 C1 = 1 = Ā B C. (ii) tr C 1 = c an tr C = x both belong to F q \ F q, as c an b = x both belong to F q \ F q. Hence, C1 an C o not belong to PSL(, q). Moreover, Ā 1 an B 1 o not commute an Ā an B o not commute. Therefore by [15, Theorem 4], Ā1, B 1 = PSL(, q ) = Ā, B. (iii) The characteristic polynomial of C 1 is λ ( c ) + 1, an its iscriminant equals c (c 4), which is a square in F q, thus C 1 is split an so its orer ivies q 1. Similarly, the characteristic polynomial of C is λ ( b ) + 1, an its iscriminant equals b (b 4), which is a non-square in F q, thus C is non-split an so its orer ivies q +1. By Lemma 4.3, Σ(Ā1, B 1, C 1 ) Σ(Ā, B, C ) = {1}, since the orers of C1 an C are relatively prime, an Ā1 an Ā are not conjugate in PSL(, q ) Proof of Theorem E. The conitions are necessary. Assume that the group PGL(, q) amits an unmixe Beauville structure of type ( (r 1, s 1, t 1 ), (r, t, s ) ). Then there exist A 1, B 1, C 1, A, B, C PGL(, q) of orers r 1, s 1, t 1, r, s, t respectively, such that A 1 B 1 C 1 = I = A B C an A 1, B 1 = PGL(, q) = A, B, implying that PGL(, q) is a quotient of the triangle groups (r 1, s 1, t 1 ) an (r, s, t ) with torsion-free kernel, an so conitions (i) an (ii) are necessary.

12 1 SHELLY GARION Therefore, we may assume that (r 1, s 1, t 1 ) an (r, s, t ) are hyperbolic an irregular, namely that they satisfy one of the Cases (α), (β), (γ) of Definition.3. If, for example, gc(r 1, r ) >, then Lemma 4.4 implies that Σ(A 1 ) Σ(A ) is non-trivial, contraicting Σ(A 1, B 1, C 1 ) Σ(A, B, C ) = {1}. Hence, conition (iii) is necessary. Since (r 1, s 1, t 1 ) an (r, s, t ) are hyperbolic an irregular, then both of them must contain at least two even numbers, one of which is greater than. Hence, we may assume that r 1, r are even an that r 1, r >. If both r 1, r ivie q 1 (resp. q + 1) then both A 1, A are split (resp. non-split) an by Lemma 4.4, Σ(A 1 ) Σ(A ) {1}, yieling a contraiction. Hence, we may assume that r 1 ivies q 1 an r ivies q + 1, an so A 1 is split an A is non-split. If one of s 1, t 1 is even an not ivies q 1, then it is necessarily an even integer greater than, thus it must ivie q + 1, an so either B 1 or C 1 is non-split. Lemma 4.4 now implies again that either Σ(B 1 ) Σ(A ) {1} or Σ(C 1 ) Σ(A ) {1}, yieling a contraiction. Hence, conition (iv) is necessary. Moreover, if either B 1 or C 1 has orer, then the above argument shows that it is necessarily split. Hence, by 3.3, if q 1 (mo 4), then Case (β) hols, an if q 3 (mo 4), then Case (γ) hols. Similarly, if either B or C has orer, then the above argument shows that it is necessarily nonsplit. Hence, by 3.3, if q 1 (mo 4), then Case (γ) hols, an if q 3 (mo 4), then Case (β) hols. Hence, conition (v) is necessary. The conitions are sufficient. Let (r 1, s 1, t 1 ) an (r, s, t ) be two triples of integers. Assume that PGL(, q) is a quotient of the triangle groups (r 1, s 1, t 1 ) an (r, s, t ) with torsion-free kernel. Then one can fin elements A 1, B 1, C 1, A, B, C PGL(, q) of orers r 1, s 1, t 1, r, s, t respectively, such that A 1 B 1 C 1 = I = A B C an A 1, B 1 = PGL(, q) = A, B, an so conitions (i) an (ii) of Definition 1.1 are fulfille. Since, by Theorem B, (r 1, s 1, t 1 ) an (r, s, t ) are hyperbolic an irregular, they must contain at least two even numbers. Hence, we may assume that r 1, r, s 1, s are even, that r 1, r >, that µ PSL (p, r 1 ) an µ PSL (p, r ) o not ivie e, an that µ PSL(p, t 1 ) an µ PSL (p, t ) both ivie e. The conition that gc(r 1, r ) now implies that one of r 1, r ivies q 1 an the other ivies q + 1. We may assume that r 1 q 1 an r q + 1, an so A 1 is split an A is non-split. Lemma 4.4 now implies that Σ(A 1 ) Σ(A ) = {1}. If s 1 is greater than, then the conition that s 1 q 1 implies that B 1 is split, an if s 1 = then Case (γ) hols, an so q 3 (mo 4), thus again B 1 is split. Lemma 4.4 implies again that Σ(B 1 ) Σ(A ) = {1}. If s is greater than, then the conition that s q + 1 implies that B is non-split, an if s = then Case (γ) hols, an so q 1 (mo 4), thus again B is non-split. Lemma 4.4 implies again that Σ(A 1 ) Σ(B ) = {1} an Σ(B 1 ) Σ(B ) = {1}. If t 1 is even an greater than, then the conition that t 1 q 1 implies that C 1 is split, an if t 1 = then Case (β) hols, an so q 1 (mo 4), thus again C 1 is split. Lemma 4.4 implies again that Σ(C 1 ) Σ(A ) = {1} an Σ(C 1 ) Σ(B ) = {1}. If t 1 is o, then necessarily gc(t 1, r ) = 1

13 ON BEAUVILLE STRUCTURES FOR PSL(, q) 13 an gc(t 1, s ) = 1, an Lemma 4.4 implies that Σ(C 1 ) Σ(A ) = {1} an Σ(C 1 ) Σ(B ) = {1}. Similarly, if t is even an greater than, then the conition that t q + 1 implies that C is non-split, an if t = then Case (β) hols, an so q 3 (mo 4), thus again C is non-split. Lemma 4.4 implies again that Σ(A 1 ) Σ(C ) = {1} an Σ(B 1 ) Σ(C ) = {1}. If t is o, then necessarily gc(r 1, t ) = 1 an gc(s 1, t ) = 1, an Lemma 4.4 implies that Σ(A 1 ) Σ(C ) = {1} an Σ(B 1 ) Σ(C ) = {1}. Moreover, either gc(t 1, t ) = 1, or gc(t 1, t ) = an C 1 is split while C is non-split, an so, by Lemma 4.4, Σ(C 1 ) Σ(C ) = {1}. To conclue, Σ(A 1, B 1, C 1 ) Σ(A, B, C ) = {1}, hence conition (iii) of Definition 1.1 is fulfille. References [1] I. Bauer, F. Catanese, Some new surfaces with p g = q = 0. Proceeing of the Fano Conference. Torino (00), [] I. Bauer, F. Catanese, F. Grunewal, Beauville surfaces without real structures. In: Geometric methos in algebra an number theory, Progr. Math., vol 35, Birkhäuser Boston, (005), 1 4. [3] A. Beauville, Surfaces Algébriques Complex. Astérisque 54, Paris (1978). [4] F. Catanese, Fibre surfaces, varieties isogenous to a prouct an relate mouli spaces. Amer. J. Math. 1, (000), [5] M.D.E. Coner, Generators for alternating an symmetric groups, J. Lonon Math. Soc. (1980) [6] M.D.E. Coner, Hurwitz groups: a brief survey, Bull. Amer. Math. Soc. 3 (1990) [7] L. E. Dickson. Linear groups with an exposition of the Galois fiel theory (Teubner, 1901). [8] B. Everitt, Alternating quotients of Fuchsian groups, J. Algebra 3 (000) [9] S. Garion, M. Penegini, New Beauville surfaces, mouli spaces an finite groups, preprint availiable at arxiv: [10] Y. Fuertes, G. González-Diez, On Beauville structures on the groups S n an A n, preprint. [11] Y. Fuertes, G. Jones, Beauville surfaces an finite groups, preprint availiable at arxiv: [1] U. Langer, G. Rosenberger, Erzeugene enlicher projektiver linearer Gruppen, Results Math. 15 (1989), no. 1-, [13] F. Levin, G. Rosenberger, Generators of finite projective linear groups. II., Results Math. 17 (1990), no. 1-, [14] M.W. Liebeck, A. Shalev, Fuchsian groups, coverings of Riemann surfaces, subgroup growth, ranom quotients an ranom walks. J. Algebra 76 (004) [15] A. M. Macbeath, Generators of the linear fractional groups, Number Theory (Proc. Sympos. Pure Math., Vol. XII, Houston, Tex., 1967), Amer. Math. Soc., Provience, R.I. (1969) [16] C. Marion, Triangle groups an PSL (q), J. Group Theory 1 (009), [17] C. Marion, On triangle generation of finite groups of Lie type, to appear in J. Group Theory. [18] L. Di Martino, M.C. Tamburini, A.E. Zalesskii. On Hurwitz groups of low rank, Comm. Algebra 8 (000), no. 11, [19] M. Suzuki, Group Theory I, Springer-Verlag, Berlin, 198. [0] M.C. Tamburini, M. Vsemirnov, Irreucible (, 3, 7)-subgroups of PGL n(f ) for n 7, J. Algebra 300 No. 1 (006), ,

14 14 SHELLY GARION [1] M.C. Tamburini, M. Vsemirnov, Hurwitz groups an Hurwitz generation, Hanbook of Algebra, vol. 4, eite by M. Hazewinkel, Elsevier (006), [] M.C. Tamburini, A.E. Zalesskii. Classical groups in imension 5 which are Hurwitz, Finite groups 003, , Walter e Gruyter, Berlin, 004. [3] R. Vincent an A. E. Zalesski, Non-Hurwitz classical groups, LMS J. Comput. Math. 10 (007), 1 8. Shelly Garion, Max-Planck-Institute for Mathematics, D Bonn, Germany aress: shellyg@mpim-bonn.mpg.e