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By George Kempf

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Let us represent these generators by contours th at do not intersect each other out of p. The preimages of these contours cut from A a fundamental domain having a vertex at q. The sequential set ipa(v) leads to a similar domain M a. Hence there is a homeomorphism A: A —>A such that A(M ) = M a and A T ^A -1 = T^. It induces an autohomeomorphism Modp , which is obviously a monomorphism. 1. □ 1. 2. Let E T. Then Mod^ = Mod Proof.

7). Let n = g + k + m and suppose 6g + 3fc + 2m > 6. Denote by T gtk>m the set of monomorphisms ip: v —* Aut(A) such that ip(v) is a sequential set of type (g, k, m). For d G 7g>n and D € Aut(A) let us set Cj Tgjstrn{d) D) = {ip G f gjktm | ip(d) = D{. Suppose k + m > 1. Consider the embedding < P : 7s,n-i >7s,n> uniquely determined by the requirements 4>(i/)> < K % + i) = C n -iC n , ^ ( c i ) = c ^ ^ 'i j C j - i C n - i C n (i = g + 2, . . , n - 1). Consider also the embedding n determined by the conditions H cl) = cn lcn - 1,

1 In other cases we set S = 1. The set of holes cg+1 , . . , cg+k decomposes into two subsets, Do - {Ci | u(ci) = 0} and Di = {a\ cv(ci) = 1}. Denote by ka = ka(P,u>) the cardinality of Da. Similarly, the punctures cg+k+1, • • •, Cn are separated into two subsets of the form Ea = {ci\ u (a ) - cc}. Let m a (a = 0,1) denote the cardinality of E a. The type of dn A rf function (P, w) is the set (g, 5, ko, k\, mo, m i). Set na = ka + m a. 1. A set (g,6,ka,m a) is o type of dn A rf function if and only if (1) ki + mi = 0 (mod 2); (2) 6 = 0 for 7ii > 0.

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