Spectral Analysis of the Adjacency Matrices for Alternating Quotients of Hyperbolic Triangle Group <inline-formula><math display="inline"><semantics><mrow><msup><mo>▵</mo><mo>*</mo></msup></mrow></semantics></math></inline-formula>(3,<i>q</i>,<i>r</i>) for <i>q</i> < <i>r</i> Primes

Abstract

Hyperbolic triangle groups are found within the category of finitely generated groups. These are topological groups formed by the reflections along the sides of a hyperbolic triangle and acting properly discontinuously on the hyperbolic plane. Higman raised a question about the simplicity of finitely generated groups. The best known example of a simple group is the alternating group An, where n≥5. This article establishes a relation between the hyperbolic triangle group denoted as ▵*(3,7,r) and the alternating group. The approach involves employing coset diagrams to establish this connection. The construction of adjacency matrices for these coset diagrams is performed, followed by a detailed examination of their spectral characteristics.

Keywords

• hyperbolic triangle group
• alternating group
• coset diagram
• adjacency matrices