On certain permutation representations of mapping class groups

Abstract

(a 1, bl, ..., ag, bg; (al, bl)'..(ag, bg)= 1), where (al, bl)=ai bi ai-lbi -1. The mapping class group of the surface, Mg, has an algebraic characterization as a subgroup of the outer automorphism group of Hg. (Out (Hg) = Aut (Hg)/Inn (Hg).) Only those automorphism classes which are induced by free automorphisms on at,..., bg mapping (al, bl)..-(ag, bg) to a conjugate of itself rather than its inverse are permitted. In [2, p. 67] the question of whether Mg/M'g is trivial or of order 2 for g>2 is raised and the study of certain representations of Mg is suggested. In this paper we study some of these representations which we now describe. Let G and Q be arbitrary groups. We call N a Q-defining subgroup of G if N< G and G/N_~ Q. Aut (G) acts as a group of permutations on all Q-defining subgroups of G. Since Inn(G) acts trivially, Out(G) acts as a group of permutations on Q-defining subgroups of G/If, further, G is finitely generated and Q is finite we obtain a homomorphism from Out(G) to some 27 k (the symmetric group on k symbols). Letting G=Hg, we obtain permutation representations of MgcOUt(Hg). It is easily seen that if we choose for Q an abelian group, the representation obtained factors through the well known representation p: Mg--,PSp2g(7Z), the projective symplectic group, obtained by considering the action of Aut(Hg) on Hg/H'g and then factoring by the center. This is proved in Lemma 2 for the case Q=2~,, where it is shown that the action of Mg on 7Zn-defining subgroups is PSp2g(7Zn) as a permutation group on points of projective (2g-1)-space over 2g,. The main result of this paper is a description of the representation when Q =Dp, the dihedral group of order 2p, p an odd prime. The result holds true for De, where q is any square free odd integer, but for simplicity we limit our discussion to the case of odd primes. We will prove the following theorem. Theorem. The action of Mg on Dp-defining subgroups of Hg, for p an odd prime, is isomorphic to the full wreath product