FIBONACCI LENGTH AND SPECIAL AUTOMORPHISMS OF FINITE (l,m n ,k)-GROUPS

We study here the Fibonacci length of finitely presented and parametric groups (b,a | a^l = b^m , (ab)^n = (ab^-1)^k) for positive integers l, m,n and k. They are indeed extensions of (l, m|n, k)-groups of M. Edjvet and R.M. Thomas considered for their finiteness property in 1997. We prove that there are some subclasses of these groups which are non-isomorphc groups of the same Fibonacci length. More interesting result is that, these lengths are independent of one of the involved parameters of the groups, and also the lengths involve the Wall number k(n). Moreover, the Fibonacci lengths of two homomorphic images of the groups have been calculated and compared with those of the groups.