On transversal group topologies

Two non-discrete Hausdorff group topologies τ 1 , τ 2 on a group G are called transversal if the least upper bound τ 1 ∨ τ 2 of τ 1 and τ 2 is the discrete topology. We show that an infinite totally bounded topological group never admits a transversal group topology and we obtain a new criterion for precompactness in lattice-theoretical terms (the existence of transversal group topologies) for a large class of Abelian groups containing all divisible and all finitely generated groups. A full description is given of the class M of all abstract Abelian groups G where this criterion is valid (i.e., when the non-precompact group topologies on G are precisely the transversable ones). It turns out that M is the class of groups G for which the submaximal topology is precompact. o give a complete description of the structure of the transversable locally compact Abelian groups.