Free Products in Prime Characteristic: A Representation of Fuchsian Groups

✍ Scribed by Stephen D. Cohen; A.M.W. Glass

• Publisher: Elsevier Science
• Year: 1996
• Tongue: English
• Weight: 265 KB
• Volume: 179
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.1996.0004

✦ Synopsis

In 1988, S. White proved by means of field theory supplemented by a geometric d Ž . argument that the real bijections x ¬ x q 1 and x ¬ x d an odd prime generate a free group of rank 2. When these maps are considered in prime Ž . characteristic p so that x ¬ x q 1 generates a cyclic group of order p 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 characteristic p, further algebraic considerations are required to establish a key polynomial lemma. Ž . By these means we obtain an analogue of White's theorem for certain countably Ž . infinite subfields L of the algebraic closure of the finite prime field GF p . For Ž . Ž .

d any odd prime d, not a divisor of p p y 1 , the maps x ¬ x q 1 and x ¬ x generate a group of bijections of such a field L that is isomorphic to the free Ž . product ‫)ޚ‬ ‫ޚ‬rp‫ޚ‬ . This implies an explicit natural algebraic faithful representation of the free product as a transitive permutation group on a countable set.