Abstract
We examine the generators of linear groups associated with the roots of Fibonacci polynomials, matrix representations of the generators, some elements, and group structures. From the viewpoint of the roots of Fibonacci polynomials, we obtain the features of various classes of linear groups under certain circumstances. Furthermore, we analyze the relationships between the roots of Fibonacci polynomials and the new generating matrices. We get some amazing relationships between the roots of Fibonacci polynomials and the extended modular group, the extended Hecke groups, and the extended generalized Hecke groups with geometric viewpoints.
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