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 < omega} of G into open sets, there is a neighborhood V of 1 such that for every x is an element of G, {n < omega: (xV) boolean AND U-n not equal empty set} is finite. We show that assuming MA, for every n is an element of 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 = circle plus(omega) 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. (C) 2013 Elsevier B.V. All rights reserved.