Transversal group topologies on non-abelian groups

Two non-discrete Hausdorff group topologies τ1, τ2 on a group G are called transversal if the least upper bound τ1∨τ2 of τ1 and τ2 is the discrete topology. We give a complete description of the transversable locally compact groups in the case they are connected (earlier, the authors gave such a description in the abelian case). In particular, a connected Lie group is transversable if and only if its center is not compact.