Certain balanced groups and 3-manifolds

Five series of groups with finite presentations are constructed. Their definition is based on the construction of some closed, compact, orientable 3-manifolds, so that these groups are balanced. The derived quotients of the groups are described. Almost all these groups are proved to be infinite; moreover, the linear groups $\operatorname {SL}(2,F)$ with $|F:{\mathbb Q}|\leqslant 6$ are involved in many of them. The relevant arguments are elementary, but the results obtained on balanced groups will be useful in further studies of 3-manifolds.