The Free Groups
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In many applications of group theory, and specifically in our subsequent analysis of the fundamental groups of the complementary spaces of knots, the groups are described by “defining relations,” or, as we are going to say later, are “presented”. We have here another (and completely different) analogy with analytic geometry. In analytic geometry a coördinate system is selected, and the geometric configuration to be studied is defined by a set of one or more equations. In the theory of group presentations the rôle that is played in analytic geometry by a coördinate system is played by a free group. Therefore, the study of group presentations must begin with a careful description of the free groups.