Partial proof of Graham Higman's conjecture related to coset diagrams

Document Type: Research Paper

Authors

1 Vice Chancellor, The Islamia University of Bahawalpur, Pakistan.

2 Department of Mathematics, Govt. Post Graduate College Jauharabad, Pakistan.

Abstract

Graham Higman has defined coset diagrams for PSL(2,ℤ). These diagrams are composed of fragments, and the fragments are further composed of two or more circuits. Q. Mushtaq has proved in 1983 that existence of a certain fragment γ of a coset diagram in a coset diagram is a polynomial f in ℤ[z]. Higman has conjectured that, the polynomials related to the fragments are monic and for a fixed degree, there are finite number of such polynomials. In this paper, we consider a family Ϝ of fragments such that each fragment in Ϝ contains one vertex fixed by
F_v [(〖xy〗^(-1) )^(s_1 ) (xy)^(s_2 ) (〖xy〗^(-1) )^(s_3 ),(xy)^(q_1 ) (〖xy〗^(-1) )^(q_2 ) (xy)^(q_3 ) ]
where s₁,s₂,s₃,q₁,q₂,q₃∈ℤ⁺, and prove Higman's conjecture for the polynomials obtained from the fragments in Ϝ.

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Main Subjects


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Volume 42, Issue 2
March and April 2016
Pages 353-369
  • Receive Date: 01 December 2013
  • Revise Date: 19 January 2015
  • Accept Date: 19 January 2015