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Coset diagrams; januarials; triangle groups; Hecke groups and genus.

Graham Higman introduced the concept of januarial as a specific map having two equal sized faces under the action of $\langle x, y : x^2 = y^k = (xy)^l = 1\rangle$ on a finite set. In this paper we take up the question posed by Graham Higman that what is the maximum number of circuits in the subgraph of a simple januarial for any value of $k$? We describe conditions under which januarials are connected and larger januarials are obtained. In an effort to look at topological features of the connected januarial, we find out genus of the januarial, genera of the two faces and number of circuits.

S MEHWISH¹ Q MUSHTAQ¹

1. Department of Mathematics, The Islamia University of Bahawalpur, Bahawalpur 63100, Pakistan

Published

18 November 2020

Final version published online

18 November 2020