Periodic Reporting for period 1 - ADA (Automata, Dynamics and Actions)
Reporting period: 2023-10-01 to 2026-03-31
This project lies at the nexus of complex and symbolic dynamics, group theory, decision problems and computation. It aims to solve major problems in each of these fields by means of automatic actions and relations.
Finite state automata, pervasive in theoretical computer science, will serve to define self-similar mathematical objects, and produce efficient algorithms to manipulate them. — I will explore a novel notion of automatically acting group, encompassing the previously unrelated notions of automatic groups, automata groups and substitutive shifts.
Geometric group theory propounds the vision of groups as geometric objects. A basic notion is volume growth, and Milnor's still open “gap problem” asks for its possible range. — In this proposal, I will give candidates of groups with very slow superpolynomial growth, defined by their automatic action on dynamical systems, and a proof strategy.
A celebrated open problem by Gromov asks whether all groups are “sofic”. This property has too many valuable consequences to always be true, yet there is no known non-example! — I will present a strategy of producing non-sofic groups closely associated to automata.
Rational maps on the Riemann sphere provide a rich supply of dynamical systems. A fundamental goal is to give a combinatorial description of the dynamics across families of maps, constructing models of parameter space. — I will encode the maps via automatic actions, and study relations between automata to produce such models. I aim to achieve a full topological description (including the long-open connectedness problem) of Milnor's “slices” of quadratic maps.
This project will tackle these fundamental questions from group theory and dynamics, and develop presently unexplored interactions between them, through a unified use of automata. It will prove decidability of certain algorithmic problems such as Dehn's and Tarski's, and construct efficient tools to further our exploration of these mathematical universes.
Finite state automata, pervasive in theoretical computer science, will serve to define self-similar mathematical objects, and produce efficient algorithms to manipulate them. — I will explore a novel notion of automatically acting group, encompassing the previously unrelated notions of automatic groups, automata groups and substitutive shifts.
Geometric group theory propounds the vision of groups as geometric objects. A basic notion is volume growth, and Milnor's still open “gap problem” asks for its possible range. — In this proposal, I will give candidates of groups with very slow superpolynomial growth, defined by their automatic action on dynamical systems, and a proof strategy.
A celebrated open problem by Gromov asks whether all groups are “sofic”. This property has too many valuable consequences to always be true, yet there is no known non-example! — I will present a strategy of producing non-sofic groups closely associated to automata.
Rational maps on the Riemann sphere provide a rich supply of dynamical systems. A fundamental goal is to give a combinatorial description of the dynamics across families of maps, constructing models of parameter space. — I will encode the maps via automatic actions, and study relations between automata to produce such models. I aim to achieve a full topological description (including the long-open connectedness problem) of Milnor's “slices” of quadratic maps.
This project will tackle these fundamental questions from group theory and dynamics, and develop presently unexplored interactions between them, through a unified use of automata. It will prove decidability of certain algorithmic problems such as Dehn's and Tarski's, and construct efficient tools to further our exploration of these mathematical universes.
In collaboration with PD1 (Ivan Mitrofanov) and PD2 (Ruiwen Dong), we have developed the theory of substitutional subshifts from the perspective of Büchi automata. In particular, we obtained a new formulation of the Pisot conjecture.
The Pisot conjecture concerns substitutional subshifts, those that are the main object of study of section a.3. It asserts that these systems are almost 1:1 extensions of a Kronecker system (translation on a compact abelian group). Preliminary work by the PI showed that substitutional subshifts are examples of automatic actions, and more precisely bounded automatic actions. With PD1 and PD2, we proved that there exists a dual action of $\mathbb Z^{d-1}$, which is also automatic; and the Pisot conjecture is equivalent to the claim that the dual action is “subexponentially bounded”, namely that its activity growth is subexponentially smaller than the growth of the subshift.
This opens a new line of attack to the conjecture, and a clear path towards its solution.
In standalone work, PD2 focused his research on decision problems in infinite groups, in particular metabelian groups, as well as their connections with algorithmic algebra and automata theory.
On one hand, he worked on equations in groups. His results include the construction of an abelian-by-cyclic group where finding solutions to quadratic equations is undecidable. He also showed undecidability of the Diophantine problem in the wreath product of two infinite cyclic groups. Both results appeared in the Proceedings of the 2025 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), see https://epubs.siam.org/doi/10.1137/1.9781611978322.59(opens in new window)
On the other hand, he worked on the Submonoid Membership problem. He showed that Submonoid Membership is decidable in quotients of n-dimensional lamplighter groups $(\mathbb{Z}/p) \wr \mathbb{Z}^n$ for prime $p$ and positive integer $n$. As a corollary, one obtains that decidability of Submonoid Membership is not stable under finite group extension. A preprint of this result is available at https://arxiv.org/abs/2409.07077(opens in new window)
The outcomes are prepublications and publications.
The Pisot conjecture concerns substitutional subshifts, those that are the main object of study of section a.3. It asserts that these systems are almost 1:1 extensions of a Kronecker system (translation on a compact abelian group). Preliminary work by the PI showed that substitutional subshifts are examples of automatic actions, and more precisely bounded automatic actions. With PD1 and PD2, we proved that there exists a dual action of $\mathbb Z^{d-1}$, which is also automatic; and the Pisot conjecture is equivalent to the claim that the dual action is “subexponentially bounded”, namely that its activity growth is subexponentially smaller than the growth of the subshift.
This opens a new line of attack to the conjecture, and a clear path towards its solution.
In standalone work, PD2 focused his research on decision problems in infinite groups, in particular metabelian groups, as well as their connections with algorithmic algebra and automata theory.
On one hand, he worked on equations in groups. His results include the construction of an abelian-by-cyclic group where finding solutions to quadratic equations is undecidable. He also showed undecidability of the Diophantine problem in the wreath product of two infinite cyclic groups. Both results appeared in the Proceedings of the 2025 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), see https://epubs.siam.org/doi/10.1137/1.9781611978322.59(opens in new window)
On the other hand, he worked on the Submonoid Membership problem. He showed that Submonoid Membership is decidable in quotients of n-dimensional lamplighter groups $(\mathbb{Z}/p) \wr \mathbb{Z}^n$ for prime $p$ and positive integer $n$. As a corollary, one obtains that decidability of Submonoid Membership is not stable under finite group extension. A preprint of this result is available at https://arxiv.org/abs/2409.07077(opens in new window)
The outcomes are prepublications and publications.
The impact of the project is nothing less than a paradigm shift: examples and notions are not anymore studied individually in their specific aspects, but rather through the all-encompassing theory of automata and regular languages.
An important step has been achieved in understanding the first-order theory of actions, and its connections to fundamental conjectures; but more work will be needed to achieve complete success.
An important step has been achieved in understanding the first-order theory of actions, and its connections to fundamental conjectures; but more work will be needed to achieve complete success.