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Presenting groups in a formal, abstract algebraic manner is both useful and powerful, yet it avoids a fascinating geometric perspective on group theory - which is also useful and powerful, particularly in the study of infinite groups. This book presents the modern, geometric approach to group theory, in an accessible and engaging approach to the subject. Topics include group actions, the construction of Cayley graphs, and connections to formal language theory and geometry. Theorems are balanced by specific examples such as Baumslag-Solitar groups, the Lamplighter group and Thompson's group. Only exposure to undergraduate-level abstract algebra is presumed, and from that base the core techniques and theorems are developed and recent research is explored. Exercises and figures throughout the text encourage the development of geometric intuition. Ideal for advanced undergraduates looking to deepen their understanding of groups, this book will also be of interest to graduate students and researchers as a gentle introduction to geometric group theory.

Infinite groups --- Groupes infinis --- Infinite groups. --- 512.54 --- Groups. Group theory --- 512.54 Groups. Group theory --- Groups, Infinite --- Group theory --- Group theory. --- Groups, Theory of --- Substitutions (Mathematics) --- Algebra

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Subgroup growth studies the distribution of subgroups of finite index in a group as a function of the index. In the last two decades this topic has developed into one of the most active areas of research in infinite group theory; this book is a systematic and comprehensive account of the substantial theory which has emerged. As well as determining the range of possible "growth types", for finitely generated groups in general and for groups in particular classes such as linear groups, a main focus of the book is on the tight connection between the subgroup growth of a group and its algebraic structure. For example the so-called PSG Theorem, proved in Chapter 5, characterizes the groups of polynomial subgroup growth as those which are virtually soluble of finite rank. A key element in the proof is the growth of congruence subgroups in arithmetic groups, a new kind of "non-commutative arithmetic", with applications to the study of lattices in Lie groups. Another kind of non-commutative arithmetic arises with the introduction of subgroup-counting zeta functions; these fascinating and mysterious zeta functions have remarkable applications both to the "arithmetic of subgroup growth" and to the classification of finite p-groups. A wide range of mathematical disciplines play a significant role in this work: as well as various aspects of infinite group theory, these include finite simple groups and permutation groups, profinite groups, arithmetic groups and strong approximation, algebraic and analytic number theory, probability, and p-adic model theory. Relevant aspects of such topics are explained in self-contained "windows", making the book accessible to a wide mathematical readership. The book concludes with over 60 challenging open problems that will stimulate further research in this rapidly growing subject.

Subgroup growth (Mathematics) --- Infinite groups. --- Croissance de sous-groupes (Mathématiques) --- Groupes infinis --- Infinite groups --- 512.54 --- Groups. Group theory --- 512.54 Groups. Group theory --- Croissance de sous-groupes (Mathématiques) --- Growth, Subgroup (Mathematics) --- Group theory --- Groups, Infinite --- Algebra. --- Group theory. --- Number theory. --- Science, Humanities and Social Sciences, multidisciplinary. --- Group Theory and Generalizations. --- Number Theory. --- Number study --- Numbers, Theory of --- Algebra --- Groups, Theory of --- Substitutions (Mathematics) --- Mathematics --- Mathematical analysis