Our aim here is to present, within the limited scope of the author’s knowledge, the state of the art of random groups. The decision to write such a survey arose from consideration of the rapidly growing number of publications on the subject, which, from a bunch of theorems, is slowly shaping into a theory. A whole section has been devoted to the statement of open problems of various difficulty. The goal was not to track down the origins of the generic way of thinking in group theory or neighboring fields, but to review the results in that branch of mathematics which treats of the groups obtained from random presentations. In particular, and mainly because of the author’s incompetence on these matters, the asymptotic theory of finite groups and properties of random elements therein are not covered. The information presented here has deliberately been limited to the works available to the author as of January 31 st, 2005, except for bibliographical references to then unpublished manuscripts, which have been updated for the reader’s convenience. I. Models of typical groups 3 I.1. Forerunners: few-relator models.......................
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