A convergence group is a group endowed with a (sequential) convergence structure that is compatible with the group operations. The convergence structure is coarse if there is no strictly larger ...convergence structure for ▫$G$▫. The authors investigate properties of abelian convergence groups that contain a dense subgroup which is coarse. Let ▫$G'$▫ be a coarse convergence group and let ▫$G$▫ be a dense subgroup of ▫$G'$▫. It is shown that ▫$G$▫ is coarse if and only if each nontrivial subgroup of ▫$G'$▫ intersects ▫$G$▫ in a nontrivial subgroup. This result is used to give conditions under which coarseness is preserved by products. The relationship between coarseness and completeness is investigated. It is shown that divisible coarse convergence groups are complete. Some examples are discussed.
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