One-Relator Products of Cyclic Groups and Fibonacci-Like Sequences

Abstract

In this paper, we mention some properties of certain generalized Fibonacci sequences we have looked at while investigating one-relator products of cyclic groups. The particular groups we have investigated are those defined by presentations of the form

$$ < a, b:{a^2} = {b^n} = R\left( {a, b} \right) = 1 > $$

, where R(a, b) is a word of the form ab^{j (1)}ab^{j (2)}…ab^{j (r)} with r ≥ 2 and 0<j(i)<n for each i. Such a group is called a one-relator product of the cyclic groups C₂ and C_n of orders 2 and n respectively, in that it is formed from the free product of C₂ and C_n by imposing the single extra relation R(a, b) = 1. We denote this group by G(n; j(1), j(2),…, j(r)).