This book is intended as an introduction, demanding a minimum of background, to some of the central ideas in the theory of groups and in geometry. It grew out of a course, for advanced undergraduates and beginning graduate students, given several times at the University of Michigan and, in 1980-81, at the University de Picardie. It is assumed that the reader has some acquaintance with the algebra of the complex plane, with analytic geometry, and with the basic concepts of linear algebra. No technical knowledge of geometry is assumed, and no knowledge of group theory, although some exposure to the fundamental ideas of group theory would probably prove helpful.

We exploit the well known close connections between group theory and geometry to develop the two subjects in parallel. Group theory is used to clarify and unify the geometry, while the geometry provides concrete and intuitive examples of groups. This has some influence on our emphasis, which is primarily combinatorial. The groups are mainly infinite groups, often given by generators and relations. The geometry is mainly incidence geometry; apart from linear algebra, we have used analytic and metric methods sparingly. In the interest of intuition we have, with one exception, confined attention to two-dimensional geometry.

Except in connection with projective geometry, we have paid little attention to axioms. We feel that this is no real loss of rigor, since, where intuition is not found sufficient, the reader can always fall back on analytic geometry to verify elementary assertions.