Permutation Representations of the Symmetry Groups of Regular Hyperbolic Tessellations

Higman has questioned which discrete hyperbolic groups [p, q] have representations onto almost all symmetric and alternating groups. We call this property 3tf and show that, except perhaps for finitely many values of/? and q, [p,q] has property JC. It is well known that the modular group F = (x,y\ x = y = 1> has the property that every alternating and symmetric group is a homomorphic image of F except A6, A7, AH, S5, S7, or S6 [5]. Higman has questioned which discrete reflective hyperbolic groups also exhibit this type of behavior. Let G be an infinite, finitely presented group. We say that G has property J^f if there is an integer N > 1 such that either An or Sn is a homomorphic image of G for all n> N. If/? and q satisfy (p — 2)(q — 2) > 4, then {p, q) denotes the tessellation of the hyperbolic plane by regular /?-gons, q meeting at each vertex, and [p, q] denotes the symmetry group of that tessellation. It is an infinite Coxeter group generated by the reflections in the sides of the right hyperbolic triangle forming the fundamental region of the tessellation, and has the presentation [p,q] = . Let us assume without loss of generality that/? ^ q. For [p, q] we shall actually use the presentation [p,q] = , (1) with the correspondence given by x = RiR2,y = R2^z and t = R2, which exhibits [p, q] as a semi-direct product [p,g] = *O\t = i>. Geometrically, x and y are rotations at the vertices of the fundamental region having angles TT/2 and nip respectively, and t is the reflection on the side joining them; [p,q] denotes the index 2 subgroup of orientation preserving isometries in [p,q]. For more details see [4]. It has been shown that [3, q] has property Jiff for all q > 6 (see [1]), and that [4, q] has property 3/C for all q > 6 (see [6]). We shall show that [p, q] has property tf for all but perhaps finitely many values of/? and q. Presumably, [p,q] has property Jf for all values of/? and q, as a careful examination of any particular case has so far led to the conclusion that it has property Jf. In view of the results cited above, and since [/?, q] is a homomorphic image of [pr, q], it is enough to show this for p prime, /? > 3. In particular, we shall show that the following theorems hold. Received 19 November 1991. 1991 Mathematics Subject Classification 20G05. The first author was supported by a grant from the Fulbright Foundation. J. London Math. Soc. (2) 48 (1993) 77-86 78 QAISER MUSHTAQ AND HERMAN SERVATIUS THEOREM 1. If q > 57, then either An or Sn is a homomorphic image of[5, q] for all n> 4(57) (58). THEOREM 2. Ifp is prime, p ^ 7, q > 3p + 2, then either An or Sn is a homomorphic image of[p,q]for all n > {2q){2q+ 1). THEOREM 3. Ifp is prime, 25 < p ^ q, then either Anor Sn is a homomorphic image of[p,q]for alln> [(2q)-\6)[(2q+ 1)—16]. Suppose that [p, q] has property 34?, p ^q, and define Np q to be the largest value of n such that An or Sn is not a homomorphic image of [p, q]. COROLLARY 1. Np g = O(q ).