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An accessible and panoramic account of the theory of random walks on groups and graphs, stressing the strong connections of the theory with other branches of mathematics, including geometric and combinatorial group theory, potential analysis, and theoretical computer science. This volume brings together original surveys and research-expository papers from renowned and leading experts, many of whom spoke at the workshop 'Groups, Graphs and Random Walks' celebrating the sixtieth birthday of Wolfgang Woess in Cortona, Italy. Topics include: growth and amenability of groups; Schr©œdinger operators and symbolic dynamics; ergodic theorems; Thompson's group F; Poisson boundaries; probability theory on buildings and groups of Lie type; structure trees for edge cuts in networks; and mathematical crystallography. In what is currently a fast-growing area of mathematics, this book provides an up-to-date and valuable reference for both researchers and graduate students, from which future research activities will undoubtedly stem.

Random walks (Mathematics) --- Stochastic processes --- Arithmetic groups

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Groupes algébriques linéaires. --- Linear algebraic groups --- Groupes arithmétiques --- Arithmetic groups

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Group theory --- Linear algebraic groups --- Arithmetic groups --- 51 --- Lie groups --- Algebraic groups, Linear --- Geometry, Algebraic --- Algebraic varieties --- Groups, Lie --- Lie algebras --- Symmetric spaces --- Topological groups --- Mathematics --- Arithmetic groups. --- Lie groups. --- Linear algebraic groups. --- 51 Mathematics

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Group theory. --- Homology theory. --- Groupes, Théorie des. --- Cohomologie. --- Groupes arithmétiques --- Arithmetic groups --- Cohomologie --- Groupes arithmétiques. --- Groupes, Théorie des --- Représentations de groupes

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This open access textbook presents a comprehensive treatment of the arithmetic theory of quaternion algebras and orders, a subject with applications in diverse areas of mathematics. Written to be accessible and approachable to the graduate student reader, this text collects and synthesizes results from across the literature. Numerous pathways offer explorations in many different directions, while the unified treatment makes this book an essential reference for students and researchers alike. Divided into five parts, the book begins with a basic introduction to the noncommutative algebra underlying the theory of quaternion algebras over fields, including the relationship to quadratic forms. An in-depth exploration of the arithmetic of quaternion algebras and orders follows. The third part considers analytic aspects, starting with zeta functions and then passing to an idelic approach, offering a pathway from local to global that includes strong approximation. Applications of unit groups of quaternion orders to hyperbolic geometry and low-dimensional topology follow, relating geometric and topological properties to arithmetic invariants. Arithmetic geometry completes the volume, including quaternionic aspects of modular forms, supersingular elliptic curves, and the moduli of QM abelian surfaces. Quaternion Algebras encompasses a vast wealth of knowledge at the intersection of many fields. Graduate students interested in algebra, geometry, and number theory will appreciate the many avenues and connections to be explored. Instructors will find numerous options for constructing introductory and advanced courses, while researchers will value the all-embracing treatment. Readers are assumed to have some familiarity with algebraic number theory and commutative algebra, as well as the fundamentals of linear algebra, topology, and complex analysis. More advanced topics call upon additional background, as noted, though essential concepts and motivation are recapped throughout.

Algebra --- Groups & group theory --- Number theory --- Associative Rings and Algebras --- Group Theory and Generalizations --- Number Theory --- Open Access --- Quaternions --- Quaternion algebras --- Quaternion orders --- Quaternion ideals --- Noncommutative algebra --- Quaternions and quadratic forms --- Ternary quadratic forms --- Simple algebras and involutions --- Lattices and integral quadratic forms --- Hurwitz order --- Quaternion algebras over local fields --- Quaternion algebras over global fields --- Adelic framework --- Idelic zeta functions --- Quaternions hyperbolic geometry --- Quaternions arithmetic groups --- Quaternions arithmetic geometry --- Supersingular elliptic curves --- Abelian surfaces with QM