Group topologies making every continuous homomorphic image to a compact group connected

Abstract

A topological group is minimally almost periodic (MinAP) if the only continuous homomorphism to any compact group is trivial. Dikranjan and Shakhmatov proved that if an abelian group can be equipped with a MinAP group topology, then for every m∈N the subgroup mG of G is either the trivial group or has infinite cardinality. In this paper we prove the following: if an abelian group G can be equipped with a group topology making all of its continuous homomorphic images to a compact group connected, then it admits a MinAP group topology. This condition becomes sufficient as well, as every MinAP topological group only has trivial continuous homomorphic images in compact groups.