Strongly complete almost maximal left invariant topologies on groups

Let G be a countably infinite group. A topology on G is left invariant if left translations are continuous. A left invariant topology is strongly complete if it is regular and for every partition { U n : n < ω } of G into open sets, there is a neighborhood V of 1 such that for every x ∈ G , { n < ω : ( x V ) ∩ U n ≠ ∅ } is finite. We show that assuming MA, for every n ∈ N , there is a strongly complete left invariant topology T on G with exactly n nonprincipal ultrafilters converging to 1, and in the case G = ⨁ ω Z 2 , T can be chosen to be a group topology. We also show that it is consistent with ZFC that if G can be embedded algebraically into a compact group, then there are no such topologies on G.