Special Session 37

Automata, Dynamics and Topology on Cantor spaces

Organizers: Altair Santos de Oliveira Tosti (Northern State University of Paraná, Brazil), Davide Perego (University of Geneva, Switzerland), Francesco Matucci (University of Milan-Bicocca, Italy)

MSC codes: 20F65

Description:

This special session brings together several active and interconnected lines of research centered on group actions on Cantor spaces. Such actions provide a unifying framework in which techniques and questions from automata theory, self-similarity, topological and symbolic dynamics, and geometric group theory naturally converge. Over the past decades, Cantor spaces have emerged as a particularly flexible setting for studying groups defined by combinatorial or dynamical data, while retaining strong geometric and algebraic structure. This thematic diversity aims to bring together experts to discuss current trends and issues, with the goal of expanding the collaborative network of each researcher and between Brazil and Italy. Group actions on Cantor spaces arise prominently in the study of automata groups, self-similar and contracting groups, Thompson-like groups, and topological full groups associated with minimal dynamical systems. They also play a central role in the construction and analysis of boundaries and limit spaces of groups and graphs, including those arising from iterated graph substitutions, rewriting systems, and recursive constructions. One of the strengths of this framework is its ability to bring together questions of very different flavors. On the algorithmic side, actions on Cantor spaces provide a natural environment for studying Dehn’s decision problems, such as the word problem, often through language-theoretic or automata-based methods. These problems play an important unifying role in group theory, giving a measure of the complexity of groups, as finitely presented groups with an unsolvable word problem exist by the work of Novikov, Boone and Britton. From a geometric and topological perspective, actions on Cantor spaces are closely linked to the structure of totally disconnected and locally compact groups, étale groupoids, and coarse or asymptotic invariants. We mention two groups acting on Cantor spaces that provided answers and examples to long standing questions: Grigorchuk’s self-similar group acting on a Cantor space is a group with intermediate word growth which answers negatively a question by Milnor about a polynomial-exponential dichotomy for the word growth and provides another negative example to one of the Burnside problem, being an infinite finitely generated group where every element has order a power of 2, while Thompson’s group V is the first example of an infinite finitely presented simple group. At the same time, strong connections exist with seemingly distant areas: interactions with knot and braid theory, fractal and self-similar structure. These viewpoints allow for a transfer of ideas between dynamics, topology, and algebra, often leading to new examples and counterexamples with exotic properties. Overall, the study of group actions on Cantor spaces offers a broad and coherent perspective in which algebraic, geometric, and homological properties of groups can be investigated simultaneously. By bringing together researchers working from different angles, dynamical, combinatorial, geometric, and algorithmic, this special session aims to highlight common themes, foster interaction between related approaches, and stimulate further developments in this rapidly evolving area.