This book deals with some aspects of linear techniques in combinatorial group theory having their origin in the work of Wilhelm Magnus in the 1930s. The central theme is the identification and properties of those subgroups of free groups which are induced by certain ideals of the integral group rings of free groups. This subject has been developed extensively, and the author seeks to present, in contemporary style, a systematic and comprehensive account of some of its developments. Included in the book are a solution of the Fox subgroup problem and an up-to-date development of the dimension subgroup problem. Aimed at graduate students and researchers in combinatorial group theory, the book requires a familiarity with the general terminology of free groups and group rings.
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