FREE-PRODUCTS IN PRIME CHARACTERISTIC - A REPRESENTATION OF FUCHSIAN-GROUPS

In 1988, S. White proved by means of field theory supplemented by a ge ometric argument that the real bijections x --> x + 1 and x --> x(d) ( d an odd prime) generate a free group of rank 2. When these maps are c onsidered in prime characteristic p (so that x --> x + 1 generates a c yclic group of order p) the geometric argument is no longer available. We show on the one hand that, generally, the geometry is redundant an d on the other that, in characteristic p, further algebraic considerat ions are required to establish a key polynomial lemma. By these means we obtain an analogue of White's theorem for certain (countably) infin ite subfields L. of the algebraic closure of the finite prime field GF (p). For any (odd) prime d, not a divisor of p(p - 1), the maps x --> x + 1 and x --> x(d) generate a group of bijections of such a field L that is isomorphic to the free product Z(Z/pZ). This implies an expli cit natural algebraic faithful representation of the free product as a transitive permutation group on a countable set. (C) 1996 Academic Pr ess, Inc.