Suitable Sets for Topological Groups
by
Sidney Morris
University of Wollongong

In this talk I survey some results on suitable sets for topological groups due to Karl Heinrich Hofmann (who wishes he could have been here, but sends warm regards to all and to Wis Comfort, in particular), Joan Cleary, Wis Comfort, Mikhail Tkachenko, Sergey Svetlichny, Des Robbie and me.

A subset S of a topological group G is said to be a topological generating set for G if the smallest closed subgroup containing S is G itself. If S is a singleton, then G is said to be monothetic. And the structure of monothetic locally compact (abelian) groups is well-known. In 1949 M. Kuranishi, using earlier work of H. Auerbach, showed that every semisimple Lie group is topologically generated by a two element set. Hofmann and Morris characterized those compact connected groups which are topologically generated by a two element set. And they also characterized those locally compact groups which are topologically generated by a finite set. This led to examining a generalization of generation by a finite set to generation by an infinite set which is in some sense "thin". Formally, a topological generating set S of a topological group G is said to be a suitable set if it is discrete and is closed in G \{1}. In this talk we show various classes of topological groups have suitable sets and present some open questions. Much of this is to appear in a forthcoming paper with Comfort, Svetlichny, Tkachenko, and Robbie.

Date received: May 31, 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 # caag-03.