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Exercises in cellular automata and groups
Ceccherini-Silberstein T., Coornaert M., Springer International Publishing, Cham, Switzerland, 2023. 627 pp. Type: Book (9783031103902)
Date Reviewed: Feb 6 2024

This book has been published as part of the “Springer Monographs in Mathematics” series. It complements the book Cellular automata and groups [1] and its forthcoming second edition by the same authors. The authors have also written many other mathematics books. This book and [1] are committed to the significant connections between the following four subject areas of mathematics: the theory of cellular automata, amenability and soficity, geometric group theory, and dynamical systems.

This new book comprises more than 600 amply worked-out exercises (added to the 306 exercises already suggested without solution in the previous book [1]). It likewise incorporates a fair quantity of background knowledge besides a demonstration of some important results that were published after 2010. The book is organized into eight chapters, just as in [1]. The chapters are on cellular automata, residually finite groups, surjunctive groups, amenable groups, the Garden of Eden theorem, finitely generated groups, local embeddability and sofic groups, and linear cellular automata. Each chapter commences with a wrap-up of the main definitions and outcomes contained in the corresponding chapter of [1]. Each answer is elaborate and completely self-contained to a large extent. All that is required for benefitting is touchstone undergraduate-level background knowledge of abstract algebra and general topology, in concert with outcomes as established in [1] and also the preceding exercises.

Remarks at the conclusion of the exercises furnish historic and bibliographic data, a report of associated new offshoots, and propositions for further interpretations. The subject matter includes an account of noteworthy applications to other stimulating themes, for example, Conway’s Game of Life; the Banach-Tarski paradox; Moore and Myhill’s classic Garden of Eden theorem and its generalities; group growth, sub-shifts, and algorithmic problems; and Kaplansky’s conjectures on group rings. The matter addressed by these exercises is linked with a large number of diverse areas of mathematics such as “general topology, dynamical systems, symbolic dynamics, group theory, combinatorial and geometric group theory, geometry and topology of low-dimensional manifolds, commutative and non-commutative rings, group rings, module theory, automata theory and theoretical computer science.” Everything is showcased in a refined and approachable manner.

The book will probably turn into a favorite among specialists in diverse areas of mathematics. It will likely assist them with teaching a potpourri of courses. The book should be accessible to both graduate and advanced undergraduate students. It will work for either classroom or independent use and is meant for both students and full-fledged researchers. Exercises in cellular automata and groups provides a utilitarian and exciting introduction to the discipline, in addition to being a reservoir of guidance for fostering new evolutions. It includes numerous references to the literature and a helpful index. Readers will appreciate the worked-out exercises, which serve as a guidepost for understanding the subject matter. Note that the same exercise may be solved in different ways.

Reviewer:  S. V. Nagaraj Review #: CR147702
1) Ceccherini-Silberstein, T.; Coornaert, M. Cellular automata and groups. Springer-Verlag, Berlin, Germany, 2010.
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