arXiv subjects

Nicolás Bitar

Publications and source records attributed to Nicolás Bitar.

6 recordsLinked to original sources

arXiv2026

Non-determinism in group actions and topological minimal self-joinings

We study the topological and geometrical aspects of non-determinism in group actions, generalizing the notion of non-expansive direction for abelian actions. Under suitable conditions on the acting group, we establish restrictions on the collection of non-deterministic horoballs (or half-spaces) for any action. We illustrate our results in the case of free and nilpotent groups, expanding upon known cases for abelian groups. We then apply these findings to standing problems concerning minimal self-joinings in topological dynamics, which are of independent interest.

math.DS↗

arXiv2026

A general framework for quasi-isometries in symbolic dynamics beyond groups

We introduce an algebraic structure which encodes a collection of countable graphs through a set of states, generators and relations. These structures, which we call blueprints, can capture standard algebraic objects such as groups, monoids or small categories, as well as geometric tiling spaces with finite local complexity. We provide a general framework for symbolic dynamics on blueprints under a partial monoid action, and for transferring invariants of their symbolic dynamics through quasi-isometries. In particular, we show that the undecidability of the domino problem, the existence of strongly aperiodic subshifts of finite type, and the existence of subshifts of finite type without computable points are all quasi-isometry invariants for finitely presented blueprints. As an application of this model, we show that two variants of the domino problem for geometric tilings of $\mathbb{R}^d$ are undecidable for $d \geq 2$ on any underlying tiling space with finite local complexity.

math.DS↗

arXiv2024

Self-Avoiding Walks on Cayley Graphs Through the Lens of Symbolic Dynamics

We study dynamical and computational properties of the set of bi-infinite self-avoiding walks on Cayley graphs, as well as ways to compute, approximate and bound their connective constant. To do this, we introduce the skeleton $X_{G,S}$ of a finitely generated group $G$ relative to a generating set $S$, which is a one-dimensional subshift made of configurations on $S$ that avoid all words that reduce to the identity. We provide a characterization of groups which have SFT skeletons and sofic skeletons: first, there exists a finite generating set $S$ such that $X_{G,S}$ is a subshift of finite type if and only if $G$ is a plain group; second, there exists $S$ such that $X_{G,S}$ is sofic if and only if $G$ is a plain group, $\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}$ or $\mathcal{D}_{\infty}\times\mathbb{Z}/2\mathbb{Z}$. We also characterize finitely generated torsion groups as groups whose skeletons are aperiodic. For connective constants, using graph height functions and bridges, we show that Cayley graphs of finitely generated torsion groups do not admit graph height functions, and that for groups that admit transitive graph height functions, the connective constant is equal to the growth rate of periodic points of the skeleton. Finally, we take a brief look at the set of bi-infinite geodesics and introduce an analog of the connective constant for the geodesic growth.

math.CO↗

arXiv2024

Realizability of Subgroups by Subshifts of Finite Type

We study the problem of realizing families of subgroups as the set of stabilizers of configurations from a subshift of finite type (SFT). This problem generalizes both the existence of strongly and weakly aperiodic SFTs. We show that a finitely generated normal subgroup is realizable if and only if the quotient by the subgroup admits a strongly aperiodic SFT. We also show that if a subgroup is realizable, its subgroup membership problem must be decidable. The article also contains the introduction of periodically rigid groups, which are groups for which every weakly aperiodic subshift of finite type is strongly aperiodic. We conjecture that the only finitely generated periodically rigid groups are virtually $\mathbb{Z}$ groups and torsion-free virtually $\mathbb{Z}^2$ groups. Finally, we show virtually nilpotent and polycyclic groups satisfy the conjecture.

math.DS↗

arXiv2023

Contributions to the Domino Problem: Seeding, Recurrence and Satisfiability

We study the seeded domino problem, the recurring domino problem and the $k$-SAT problem on finitely generated groups. These problems are generalization of their original versions on $\mathbb{Z}^2$ that were shown to be undecidable using the domino problem. We show that the seeded and recurring domino problems on a group are invariant under changes in the generating set, are many-one reduced from the respective problems on subgroups, and are positive equivalent to the problems on finite index subgroups. This leads to showing that the recurring domino problem is decidable for free groups. Coupled with the invariance properties, we conjecture that the only groups in which the seeded and recurring domino problems are decidable are virtually free groups. In the case of the $k$-SAT problem, we introduce a new generalization that is compatible with decision problems on finitely generated groups. We show that the subgroup membership problem many-one reduces to the $2$-SAT problem, that in certain cases the $k$-SAT problem many one reduces to the domino problem, and finally that the domino problem reduces to $3$-SAT for the class of scalable groups.

math.CO↗

arXiv2022

Strongly Aperiodic SFTs on Generalized Baumslag-Solitar groups

We look at constructions of aperiodic SFTs on fundamental groups of graph of groups. In particular we prove that all generalized Baumslag-Solitar groups (GBS) admit a strongly aperiodic SFT. Our proof is based on a structural theorem by Whyte and on two constructions of strongly aperiodic SFTs on $\mathbb{F}_n\times \mathbb{Z}$ and $BS(m,n)$ of our own. Our two constructions rely on a path-folding technique that lifts an SFT on $\mathbb{Z}^2$ inside an SFT on $\mathbb{F}_n\times \mathbb{Z}$ or an SFT on the hyperbolic plane inside an SFT on $BS(m,n)$. In the case of $\mathbb{F}_n\times \mathbb{Z}$ the path folding technique also preserves minimality, so that we get minimal strongly aperiodic SFTs on unimodular GBS groups.

cs.DM↗