A C K N O W L E D G E M E N T S The main results of this paper were obtained in the period November, 1988 - December, 1990 when the first author was a Research Fellow at the Mathematical Institute of the Bulgarian Academy of Sciences (Sofia, Bulgaria), and the second author was a Junior Research Fellow of the Faculty of Mechanics and Mathematics of Moscow State University (Moscow, USSR). During this period the first author was visiting twice the Steklov Mathematical Institute of the Soviet Academy of Sciences (Moscow, USSR), and the second author was visiting twice the Mathe- matical Institute of the Bulgarian Academy of Sciences (Sofia, Bulgaria). The final stage of the work took place in 1991/1992 when the first author was a Lecturer at the University of TAquila (FAquila, Italy), Visiting Professor in the Department of Mathematics and Statistics of York University (Toronto, Canada) and Visiting Professor in the Department of Mathematics of Padua University (Padua, Italy), and the second author was a Distinguished Visiting Associate Professor in the De- partment of Mathematics and Statistics of Miami University (Oxford, OH, U.S.A.). The authors would like to use this opportunity to thank all these institutions for their generous hospitality and support. Further thanks go to the referees for their suggestions. 1991 Mathematic s Subject Classification: Primary: 22A05, 54D30 Secondary: 03E10, 03E35, 04A10, 20E05, 20E10, 20E26, 20E34, 20F50, 20K10, 20K20, 20K45, 22C05, 54A25, 54A35, 54C25, 54D05, 54D20, 54H11. Ke y words and phrases: group, topological group, group topology, pseudocom- pact space, connected space, locally connected space, variety of groups, torsion group, free group. IX
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