Free Products in Prime Characteristic: A Representation of Fuchsian Groups

Abstract

In 1988, S. White proved by means of field theory supplemented by a geometric argument that the real bijectionsx↦x+1 andx↦xd(dan odd prime) generate a free group of rank 2. When these maps are considered in prime characteristicp(so thatx↦x+1 generates a cyclic group of orderp) the geometric argument is no longer available. We show on the one hand that, generally, the geometry is redundant and on the other that, in characteristicp, further algebraic considerations are required to establish a key polynomial lemma. By these means we obtain an analogue of White's theorem for certain (countably) infinite subfieldsLof the algebraic closure of the finite prime field GF(p). For any (odd) primed, not a divisor ofp(p−1), the mapsx↦x+1 andx↦xdgenerate a group ofbijectionsof such a fieldLthat is isomorphic to the free product Z∗(Z/pZ). This implies an explicit natural algebraic faithful representation of the free product as a transitive permutation group on a countable set.