GROUPS WITH NO FREE SUBSEMIGROUPS

We look at groups which have no (nonabelian) free subsemigroups. It is known that a finitely generated solvable group with no free subsemigr oup is nilpotent-by-finite. Conversely nilpotent-by-finite groups have no free subsemigroups. Torsion-free residually finite-p groups with n o free subsemigroups can have very complicated structure, but with som e extra condition on the subsemigroups of such a group one obtains sat isfactory results. These results are applied to ordered groups, right- ordered groups, and locally indicable groups.