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On convergence groups with dense coarse subgroups. (English) Zbl 0637.22002
A convergence group is a topological group G together with a compatible convergence structure $${\mathcal C}$$ for G - that is, a subset $${\mathcal C}$$ of $$G^{\omega}\times G$$ (the convergent sequences, with their limits) satisfying certain natural conditions; for example: (i) for $$p\in G$$ the constant function $$c_ p$$ satisfies $$<c_ p,p>\in {\mathcal C}$$; (i) if $$<s,p>\in {\mathcal C}$$ and t is a subsequence of s, then $$<t,p>\in {\mathcal C}$$; (iii) if $$<s,p>\in {\mathcal C}$$ and $$<t,q>\in {\mathcal C}$$ then $$<s-t,p- q>\in {\mathcal C}$$. A convergence group $$<G,{\mathcal C}>$$ is said to be coarse if no convergence structure $${\mathcal D}$$ for $${\mathcal G}$$ satisfies $${\mathcal C}\subseteq {\mathcal D}$$, $${\mathcal C}\neq {\mathcal D}$$. In partial analogy with the theory of minimal topological groups (which coarse convergence groups closely resemble in certain respects), the authors show that a dense subgroup G of a coarse convergence group G’ is itself coarse if and only if G meets each subgroup H of G’ such that $$| H| >1$$. It follows that (1) a coarse convergence group need not be sequentially compact and (2) the product of two coarse convergence groups, one of which is dense in a sequentially compact convergence group, is again coarse.
§ 3 deals with relations between coarseness and completeness. It is shown inter alia that every divisible Abelian group admist a (necessarily complete) non-discrete coarse convergence structure.
Reviewer: W.W.Comfort

##### MSC:
 22A05 Structure of general topological groups 54A20 Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.) 20K45 Topological methods for abelian groups
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