62 research outputs found
Tiling Problems on Baumslag-Solitar groups
We exhibit a weakly aperiodic tile set for Baumslag-Solitar groups, and prove
that the domino problem is undecidable on these groups. A consequence of our
construction is the existence of an arecursive tile set on Baumslag-Solitar
groups.Comment: In Proceedings MCU 2013, arXiv:1309.104
Realization of aperiodic subshifts and uniform densities in groups
A theorem of Gao, Jackson and Seward, originally conjectured to be false by
Glasner and Uspenskij, asserts that every countable group admits a
-coloring. A direct consequence of this result is that every countable group
has a strongly aperiodic subshift on the alphabet . In this article,
we use Lov\'asz local lemma to first give a new simple proof of said theorem,
and second to prove the existence of a -effectively closed strongly
aperiodic subshift for any finitely generated group . We also study the
problem of constructing subshifts which generalize a property of Sturmian
sequences to finitely generated groups. More precisely, a subshift over the
alphabet has uniform density if for every
configuration the density of 's in any increasing sequence of balls
converges to . We show a slightly more general result which implies
that these subshifts always exist in the case of groups of subexponential
growth.Comment: minor typos correcte
Simulation of Effective Subshifts by Two-dimensional Subshifts of Finite Type
International audienceIn this article we study how a subshift can simulate another one, where the notion of simulation is given by operations on subshifts inspired by the dynamical systems theory (factor, projective subaction...). There exists a correspondence between the notion of simulation and the set of forbidden patterns. The main result of this paper states that any effective subshift of dimension d – that is a subshift whose set of forbidden patterns can be generated by a Turing machine – can be obtained by applying dynamical operations on a subshift of finite type of dimension d + 1 – a subshift that can be defined by a finite set of forbidden patterns. This result improves Hochman's [Hoc09]
An Order on Sets of Tilings Corresponding to an Order on Languages
Traditionally a tiling is defined with a finite number of finite forbidden
patterns. We can generalize this notion considering any set of patterns.
Generalized tilings defined in this way can be studied with a dynamical point
of view, leading to the notion of subshift. In this article we establish a
correspondence between an order on subshifts based on dynamical transformations
on them and an order on languages of forbidden patterns based on computability
properties
The Domino Problem is Undecidable on Surface Groups
We show that the domino problem is undecidable on orbit graphs of non-deterministic substitutions which satisfy a technical property. As an application, we prove that the domino problem is undecidable for the fundamental group of any closed orientable surface of genus at least 2
Domino Snake Problems on Groups
In this article we study domino snake problems on finitely generated groups.
We provide general properties of these problems and introduce new tools for
their study. The first is the use of symbolic dynamics to understand the set of
all possible snakes. Using this approach we solve many variations of the
infinite snake problem including the geodesic snake problem for certain classes
of groups. Next, we introduce a notion of embedding that allows us to reduce
the decidability of snake problems from one group to another. This notion
enable us to establish the undecidability of the infinite snake and ouroboros
problems on nilpotent groups for any generating set, given that we add a
well-chosen element. Finally, we make use of monadic second order logic to
prove that domino snake problems are decidable on virtually free groups for all
generating sets.Comment: Accepted to FCT 2023. Comments welcome
On the domino problem of the Baumslag-Solitar groups
In [1] we construct aperiodic tile sets on the Baumslag-Solitar groups BS(m, n). Aperiodicity plays a central role in the undecidability of the classical domino problem on Z2, and analogously to this we state as a corollary of the main construction that the Domino problem is undecidable on all Baumslag-Solitar groups. In the present work we elaborate on the claim and provide a full proof of this fact. We also provide details of another result reported in [1]: there are tiles that tile the Baumslag-Solitar group BS(m, n)but none of the valid tilings is recursive. The proofs are based on simulating piecewise affine functions by tiles on BS(m, n).</p
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