tailieunhanh - Báo cáo toán học: "Infinitely many hypermaps of a given type and genus"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: Infinitely many hypermaps of a given type and genus. | Infinitely many hypermaps of a given type and genus Gareth A. Jones School of Mathematics University of Southampton Southampton SO17 1BJ UK ones@ Daniel Pinto CMUC Department of Mathematics University of Coimbra 3001-454 Coimbra Portugal dpinto@ Submitted Jul 20 2010 Accepted Jul 29 2010 Published Nov 5 2010 Mathematics Subject Classification 05C10 05C25 20E07 Abstract It is conjectured that given positive integers l m n with l-1 m-1 n-1 1 and an integer g 0 the triangle group A A l m n X Y Z Xl Ym Zn XYZ 1 contains infinitely many subgroups of finite index and of genus g. A slightly stronger version of this conjecture is as follows given positive integers l m n with l-1 m-1 n-1 1 and an integer g 0 there are infinitely many nonisomorphic compact orientable hypermaps of type l m n and genus g. We prove that these conjectures are true when two of the parameters l m n are equal by showing how to construct appropriate hypermaps. 1 Introduction The following conjecture arose in discussions with Jurgen Wolfart Conjecture A . Given positive integers l m n with l-1 m-1 n-1 1 and an integer g 0 the triangle group A A l m n X Y Z Xl Ym Zn ZYZ 1 contains infinitely many subgroups of finite index and of genus g. The well-known connections between triangle groups and hypermaps discussed in 4 for example yield the following slightly stronger form of this conjecture see section 3 THE ELECTRONIC JOURNAL OF COMBINATORICS 17 2010 R148 1 Conjecture B . Given positive integers l m n with l-1 m-1 n-1 1 and an integer g 0 there are infinitely many nonisomorphic compact orientable hypermaps of type l m n and of genus g. In these conjectures which are independent of the ordering of l m and n it is necessary to impose the inequality to avoid trivial cases. The natural action of A is on the Riemann sphere the complex plane or the hyperbolic plane as l-1 m-1 n-1 1 1 or 1. In the first case A is finite and there are only finitely many hypermaps of a .

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