Recent Advances in Minimal Topological Groups
by
Dikran Dikranjan
Udine University

Dedicated to the memory of Ivan Prodanov

A Hausdorff topological group (G, \tau) is minimal if \tau is a minimal element of the partially ordered (with respect to inclusion) set of Hausdorff group topologies on the group G. Introduced by R. M. Stephenson, Jr. [Math. Ann., 192 (1971) 193-195] as a natural generalization of compact groups, minimal groups turned out to be quite unpredictable. More precisely, many typical properties of the compact groups fail in general for a minimal groups: a quotient of a minimal group need not be minimal, the product of two minimal groups need not be minimal, a closed subgroup of a minimal group need not be minimal, a complete minimal group need not be compact, the character and the pseudocharacter of a minimal group need not coincide etc. This phenomenon created many hard problems and their solution developed gradually this area in the last 25 years.

The last three of the five properties listed above are present in the abelian case: Iv. Prodanov and L. Stoyanov [C. R. Acad. Bulgare Sci. 37 (1984) 23-26] established the compactness of the complete minimal abelian groups (i. e., precompactness of the minimal abelian groups). This question dominated the theory of minimal groups for a period of almost ten years. The other two properties are easy to check in the abelian case.

On the other hand, the first two properties strongly fail even in the abelian case. This justified the isolation by Iv. Prodanov [Ann. Univ. Sofia Fac. Math. Méc. 69 (1974/75) 5-11] of the smaller class of totally minimal groups characterized by this property (i. e., Hausdorff groups having all Hausdorff quotients minimal, or equivalently, satisfying the open mapping theorem). Analogously, Stoyanov [1982] introduced the perfectly minimal groups (the groups G such that G ×H is minimal for every minimal group H).

This survey intends to present some recent trends and results in minimal groups. Some of them are related to the permanence properties listed above and the precompactness theorem of Prodanov and Stoyanov. Others are related to the study of those generalizations of compactness which together with minimality yield compactness. Here is a typical result in this direction: every connected, countably compact, minimal abelian group is compact. None of the properties "connected", "countably compact", "minimal" or äbelian" can be omitted here. On the other hand, "connected and minimal" can be replaced by "totally minimal".

Date received: June 24, 1996

Copyright © 1996 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caah-17.