SOLVABILITY OF FREE PRODUCTS, CAYLEY GRAPHS AND COMPLEXES

In this paper, we verify the solvability of free product of finite cyclic groups with topological methods. We use Cayley graphs and Everitt methods to con- struct suitable 2-complexes corresponding to the presentations of groups and their commutator subgroups. In particular, using these methods, we prove that the com- mutator subgroup of Zm * Zn is free of rank (m - 1)(n - 1) for all m; n ≥ 2.