Strongly Summable Ultrafilters, Union Ultrafilters, and the Trivial Sums Property

Abstract

We answer two questions of Hindman, Stepr\=ans and Strauss, namely we prove that every strongly summable ultrafilter on an abelian group is sparse and has the trivial sums property. Moreover we show that in most cases the sparseness of the given ultrafilter is a consequence of its being isomorphic to a union ultrafilter. However, this does not happen in all cases: we also construct (assuming Martin's Axiom for countable partial orders, i.e. $\mathrm{cov}(\mathcal M)=\mathfrak c$), on the Boolean group, a strongly summable ultrafilter that is not additively isomorphic to any union ultrafilter.