## Overview

My research aims to study the rich interplay between the dynamics of group actions and computability to further understand both phenomena from a unified perspective.

A more detailed exposition can be found by clicking on my self portrait on the right.

## Publications

- Preprints
**[2020] Gibbsian representations of continuous specifications: the theorems of Kozlov and Sullivan revisited.**

with Ricardo Gómez, Brian Marcus, Tom Meyerovitch and Siamak Taati.

- File:
- [art_KozlovSullivan_BGMMT.pdf]
- Arxiv:
- https://arxiv.org/abs/2001.03880
- DOI:
- Not yet available.
- Abstract:
- The theorems of Kozlov and Sullivan characterize Gibbs measures as measures with positive continuous specifications. More precisely, Kozlov showed that every positive continuous specification on symbolic configurations of the lattice is generated by a norm-summable interaction. Sullivan showed that every shift-invariant positive continuous specification is generated by a shift-invariant interaction satisfying the weaker condition of variation-summability. These results were proven in the 1970s. An open question since that time is whether Kozlov's theorem holds in the shift-invariant setting, equivalently whether Sullivan's conclusion can be improved from variation-summability to norm-summability. We show that the answer is no: there exist shift-invariant positive continuous specifications that are not generated by any shift-invariant norm-summable interaction. On the other hand, we give a complete proof of an extension, suggested by Kozlov, of Kozlov's theorem to a characterization of positive continuous specifications on configuration spaces with arbitrary hard constraints. We also present an extended version of Sullivan's theorem. Aside from simplifying some of the arguments in the original proof, our new version of Sullivan's theorem applies in various settings not covered by the original proof. In particular, it applies when the support of the specification is the hard-core shift or the two-dimensional $q$-coloring shift for $q\geq 6$.
- Bibtex:
- @misc{Barbieri_Gomez_Marcus_Meyerovitch_Taati_2020,

Author = {Sebasti{\'{a}}n Barbieri and Ricardo G{\'{o}}mez and Brian Marcus and Tom Meyerovitch and Siamak Taati},

Title = {Gibbsian representations of continuous specifications: the theorems of {K}ozlov and {S}ullivan revisited},

Year = {2020},

Eprint = {arXiv:2001.03880},

}**[2019] Markovian properties of continuous group actions: algebraic actions, entropy and the homoclinic group.**

with Felipe García Ramos and Hanfeng Li.

- File:
- [art_markovianalgebraic_BGL.pdf]
- Arxiv:
- https://arxiv.org/abs/1911.00785
- DOI:
- Not yet available.
- Abstract:
- We provide a unifying approach which links results on algebraic actions by Lind and Schmidt, Chung and Li, and a topological result by Meyerovitch that relates entropy to the set of asymptotic pairs. In order to do this we introduce a series of Markovian properties and, under the assumption that they are satisfied, we prove several results that relate topological entropy and asymptotic pairs (the homoclinic group in the algebraic case). As new applications of our method, we give a characterization of the homoclinic group of any finitely presented expansive algebraic action of (1) any elementary amenable, torsion-free group or (2) any left orderable amenable group, using the language of independence entropy pairs.
- Bibtex:
- @misc{Barbieri_GarciaRamos_Li_2019,

Author = {Sebasti{\'{a}}n Barbieri and Felipe Garc{\'i}a-Ramos and Hanfeng Li},

Title = {Markovian properties of continuous group actions: algebraic actions, entropy and the homoclinic group},

Year = {2019},

Eprint = {arXiv:1911.00785},

}

- Journals
**[to appear] On the entropies of subshifts of finite type on countable amenable groups.***To appear in Groups, Geometry and Dynamics.*

- File:
- [art_entropia_B.pdf]
- Journal:
- Groups, Geometry and Dynamics.
- Arxiv:
- https://arxiv.org/abs/1905.10015
- DOI:
- Not yet available.
- Abstract:
- Let $G,H$ be two countable amenable groups. We introduce the notion of group charts, which gives us a tool to embed an arbitrary $H$-subshift into a $G$-subshift. Using an entropy addition formula derived from this formalism we prove that whenever $H$ is finitely presented and admits a subshift of finite type (SFT) on which $H$ acts freely, then the set of real numbers attained as topological entropies of $H$-SFTs is contained in the set of topological entropies of $G$-SFTs modulo an arbitrarily small additive constant for any finitely generated group $G$ which admits a translation-like action of $H$. In particular, we show that the set of topological entropies of $G$-SFTs on any such group which has decidable word problem and admits a translation-like action of $\mathbb{Z}^2$ coincides with the set of non-negative upper semi-computable real numbers. We use this result to give a complete characterization of the entropies of SFTs in several classes of groups.
- Bibtex:
- @misc{Barbieri_2019,

Author = {Sebasti{\'{a}}n Barbieri},

Title = {On the entropies of subshifts of finite type on countable amenable groups},

Year = {2019},

Eprint = {arXiv:1905.10015},

}**[to appear] A hierarchy of topological systems with completely positive entropy.**

with Felipe García-Ramos.*To appear in Journal d'Analyse Mathématique.*

- File:
- [art_tcpehierarchy_BG.pdf]
- Journal:
- Journal d'Analyse Mathématique.
- Arxiv:
- https://arxiv.org/abs/1803.01948
- DOI:
- Not yet available.
- Abstract:
- We define a hierarchy of systems with topological completely positive entropy in the context of countable amenable continuous group actions on compact metric spaces. For each countable ordinal we construct a dynamical system on the corresponding level of the aforementioned hierarchy and provide subshifts of finite type for the first three levels. We give necessary and sufficient conditions for entropy pairs by means of the asymptotic relation on systems with the pseudo-orbit tracing property, and thus create a bridge between a result by Pavlov and a result by Meyerovitch. As a corollary, we answer negatively an open question by Pavlov regarding necessary conditions for completely positive entropy.
- Bibtex:
- @misc{Barbieri_GarciaRamos_2018,

Author = {Sebasti{\'{a}}n Barbieri and Felipe Garc{\'{i}}a-Ramos},

Title = {A hierarchy of topological systems with completely positive entropy},

Year = {2018},

Eprint = {arXiv:1803.01948},

}**[2020] Equivalence of relative Gibbs and relative equilibrium measures for actions of countable amenable groups.**

with Ricardo Gómez, Brian Marcus and Siamak Taati.*in Nonlinearity.*

- File:
- [art_relativedlr_BGMT.pdf]
- Journal:
- Nonlinearity, 2020.
- Arxiv:
- https://arxiv.org/pdf/1809.00078
- DOI:
- 10.1088/1361-6544/ab6a75
- Abstract:
- We formulate and prove a very general relative version of the Dobrushin-Lanford-Ruelle theorem which gives conditions on constraints of configuration spaces over a finite alphabet such that for every absolutely summable relative interaction, every translation-invariant relative Gibbs measure is a relative equilibrium measure and vice versa. Neither implication is true without some assumption on the space of configurations. We note that the usual finite type condition can be relaxed to a much more general class of constraints. By "relative" we mean that both the interaction and the set of allowed configurations are determined by a random environment. The result includes many special cases that are well known. We give several applications including (1) Gibbsian properties of measures that maximize pressure among all those that project to a given measure via a topological factor map from one symbolic system to another; (2) Gibbsian properties of equilibrium measures for group shifts defined on arbitrary countable amenable groups; (3) A Gibbsian characterization of equilibrium measures in terms of equilibrium condition on lattice slices rather than on finite sets; (4) A relative extension of a theorem of Meyerovitch, who proved a version of the Lanford--Ruelle theorem which shows that every equilibrium measure on an arbitrary subshift satisfies a Gibbsian property on interchangeable patterns.
- Bibtex:
- @article{Barbieri_2020,

doi = {10.1088/1361-6544/ab6a75},

year = 2020,

month = {mar},

publisher = {{IOP} Publishing},

volume = {33},

number = {5},

pages = {2409--2454},

author = {Sebasti{\'{a}}n Barbieri and Ricardo G{\'{o}}mez and Brian Marcus and Siamak Taati},

title = {Equivalence of relative {G}ibbs and relative equilibrium measures for actions of countable amenable groups},

journal = {Nonlinearity}

}**[2019] A geometric simulation theorem on direct products of finitely generated groups.**

*In Discrete Analysis.*

- File:
- [art_simgrig_B.pdf]
- Journal:
- Discrete Analysis, 2019.
- Arxiv:
- https://arxiv.org/abs/1706.00626
- DOI:
- 10.19086/da.8820
- Abstract:
- We show that every effectively closed action of a finitely generated group $G$ on a closed subset of $\{0,1\}^{\mathbb{N}}$ can be obtained as a topological factor of the $G$-subaction of a $(G \times H_1 \times H_2)$-subshift of finite type (SFT) for any choice of infinite and finitely generated groups $H_1,H_2$. As a consequence, we obtain that every group of the form $G_1 \times G_2 \times G_3$ admits a non-empty strongly aperiodic SFT subject to the condition that each $G_i$ is finitely generated and has decidable word problem. As a corollary of this last result we prove the existence of non-empty strongly aperiodic SFT in a large class of branch groups, notably including the Grigorchuk group.
- Bibtex:
- @article{Barbieri_2019_DA,

title={A geometric simulation theorem on direct products of finitely generated groups},

doi = {10.19086/da.8820},

journal={Discrete Analysis},

author={Sebasti{\'a}n Barbieri},

number={9},

year={2019},

month = jun,

}**[2019] A generalization of the simulation theorem for semidirect products.**

with Mathieu Sablik.*In Ergodic Theory and Dynamical Systems.*

- File:
- [art_simulation_BS.pdf]
- Journal:
- Ergodic Theory and Dynamical Systems, 2019.
- Arxiv:
- https://arxiv.org/abs/1608.00357
- HAL:
- https://hal.archives-ouvertes.fr/hal-01970758v1
- DOI:
- 10.1017/etds.2018.21
- Abstract:
- We generalize a result of Hochman in two simultaneous directions: instead of realizing an arbitrary effectively closed $\mathbb{Z}^d$ action as a factor of a subaction of a $\mathbb{Z}^{d+2}$-SFT we realize an action of a finitely generated group analogously in any semidirect product of the group with $\mathbb{Z}^2$. Let $H$ be a finitely generated group and $G = \mathbb{Z}^2 \rtimes_{\varphi} H$ a semidirect product. We show that for any effectively closed $H$-dynamical system $(Y,T)$ where $Y \subset \{0,1\}^{\mathbb{N}}$, there exists a $G$-subshift of finite type $(X,\sigma)$ such that the $H$-subaction of $(X,\sigma)$ is an extension of $(Y,T)$. In the case where $T$ is an expansive action, a subshift conjugated to $(Y,T)$ can be obtained as the $H$-projective subdynamics of a sofic $G$-subshift. As a corollary, we obtain that $G$ admits a non-empty strongly aperiodic subshift of finite type whenever the word problem of $H$ is decidable.
- Bibtex:
- @article{Barbieri_Sablik_2019,

title={A generalization of the simulation theorem for semidirect products},

volume={39},

DOI={10.1017/etds.2018.21},

number={12},

journal={Ergodic Theory and Dynamical Systems},

publisher={Cambridge University Press},

author={Barbieri, Sebasti{\'{a}}n and Sablik, Mathieu},

year={2019},

pages={3185--3206}

}**[2019] Realization of aperiodic subshifts and uniform densities in groups.**

with Nathalie Aubrun and Stéphan Thomassé.*In Groups, Geometry and Dynamics.*

- File:
- [art_densities_ABT.pdf]
- Journal:
- Groups, Geometry and Dynamics, 2019.
- Arxiv:
- https://arxiv.org/abs/1507.03369
- HAL:
- https://hal.archives-ouvertes.fr/hal-01989763v1
- DOI:
- 10.4171/GGD/487
- Abstract:
- A theorem of Gao, Jackson and Seward, originally conjectured to be false by Glasner and Uspenskij, asserts that every countable group admits a 2-coloring. A direct consequence of this result is that every countable group has a strongly aperiodic subshift on the alphabet $\{0,1\}$. In this article, we use Lovász local lemma to first give a new simple proof of said theorem, and second to prove the existence of a $G$-effectively closed strongly aperiodic subshift for any finitely generated group $G$. 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 $\{0,1\}$ has uniform density $\alpha \in [0,1]$ if for every configuration the density of 1's in any increasing sequence of balls converges to $\alpha$. We show a slightly more general result which implies that these subshifts always exist in the case of groups of subexponential growth.
- Bibtex:
- @article{Aubrun_Barbieri_Thomasse_2019,

doi = {10.4171/ggd/487},

url = {https://doi.org/10.4171/ggd/487},

year = {2019},

publisher = {European Mathematical Publishing House},

volume = {13},

number = {1},

pages = {107--129},

author = {Nathalie Aubrun and Sebasti{\'{a}}n Barbieri and St{\'{e}}phan Thomass{\'{e}}},

title = {Realization of aperiodic subshifts and uniform densities in groups},

journal = {Groups, Geometry, and Dynamics}

}**[2017] A notion of effectiveness for subshifts on finitely generated groups.**

with Nathalie Aubrun and Mathieu Sablik.*In Theoretical Computer Science.*

- File:
- [art_geffective_ABS.pdf]
- Journal:
- Theoretical Computer Science, 2017.
- Arxiv:
- https://arxiv.org/abs/1412.2582
- HAL:
- https://hal.archives-ouvertes.fr/hal-01970761
- DOI:
- 10.1016/j.tcs.2016.11.033
- Abstract:
- We generalize the classical definition of effectively closed subshift to finitely generated groups. We study classical stability properties of this class and then extend this notion by allowing the usage of an oracle to the word problem of a group. This new class of subshifts forms a conjugacy class that contains all sofic subshifts. Motivated by the question of whether there exists a group where the class of sofic subshifts coincides with that of effective subshifts, we show that the inclusion is strict for several groups, including recursively presented groups with undecidable word problem, amenable groups and groups with more than two ends. We also provide an extended model of Turing machine which uses the group itself as a tape and characterizes our extended notion of effectiveness. As applications of these machines we prove that the origin constrained domino problem is undecidable for any group of the form $G\times \mathbb{Z}$ subject to a technical condition on G and we present a simulation theorem which is valid in any finitely generated group.
- Bibtex:
- @article{Aubrun_Barbieri_Sablik_2017,

title = "A notion of effectiveness for subshifts on finitely generated groups",

journal = "Theoretical Computer Science",

volume = "661",

pages = "35 - 55",

year = "2017",

issn = "0304-3975",

doi = "https://doi.org/10.1016/j.tcs.2016.11.033",

url = "http://www.sciencedirect.com/science/article/pii/S0304397516306983",

author = "Nathalie Aubrun and Sebasti{\'a}n Barbieri and Mathieu Sablik",

keywords = "Symbolic dynamics, Turing machines, Word problems, Models of computation"

}

- International Conferences
**[2019] The domino problem is undecidable on surface groups.**

with Nathalie Aubrun and Etienne Moutot.*In MFCS*

- File:
- [art_dpsurface_ABM.pdf][conf_dpsurface_ABM.pdf]
- Conference:
- In MFCS LIPIcs Volume 138, 2019.
- Arxiv:
- https://arxiv.org/abs/1811.08420 (full version)
- Proceedings:
- http://drops.dagstuhl.de/opus/volltexte/2019/10990/ (Conference proceedings)
- DOI:
- 10.4230/LIPIcs.MFCS.2019.46
- Abstract:
- 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.
- Bibtex:
- @InProceedings{Aubrun_Barbieri_Moutot_2018,

author = {Nathalie Aubrun and Sebasti{\'a}n Barbieri and Etienne Moutot},

title = {{The Domino Problem is Undecidable on Surface Groups}},

booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},

pages = {46:1--46:14},

series = {Leibniz International Proceedings in Informatics (LIPIcs)},

ISBN = {978-3-95977-117-7},

ISSN = {1868-8969},

year = {2019},

volume = {138},

editor = {Peter Rossmanith and Pinar Heggernes and Joost-Pieter Katoen},

publisher = {Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},

address = {Dagstuhl, Germany},

URL = {http://drops.dagstuhl.de/opus/volltexte/2019/10990},

URN = {urn:nbn:de:0030-drops-109900},

doi = {10.4230/LIPIcs.MFCS.2019.46},

annote = {Keywords: tilings, substitutions, SFTs, decidability, domino problem}

}**[2016] The domino problem for self-similar structures.**

with Mathieu Sablik.*In CIE.*

- File:
- [art_fractaldp_BS.pdf]
- Conference:
- Computing in Europe, 2016.
- HAL:
- https://hal.archives-ouvertes.fr/hal-01489000
- DOI:
- 10.1007/978-3-319-40189-8_21
- Abstract:
- We define the domino problem for tilings over self-similar structures of $\mathbb{Z}^d$ given by forbidden patterns. In this setting we exhibit non-trivial families of subsets with decidable and undecidable domino problem.
- Bibtex:
- @InProceedings{Barbieri_Sablik_2016,

author="Barbieri, Sebasti{\'a}n and Sablik, Mathieu",

editor="Beckmann, Arnold and Bienvenu, Laurent and Jonoska, Nata{\v{s}}a",

title="The Domino Problem for Self-similar Structures",

booktitle="Pursuit of the Universal",

year="2016",

publisher="Springer International Publishing",

address="Cham",

pages="205--214",

isbn="978-3-319-40189-8"

}**[2016] The group of reversible Turing machines.**

with Jarkko Kari and Ville Salo.*In AUTOMATA.*

- File:
- [art_grouprevtm_BKS.pdf]
- Conference:
- AUTOMATA, 2016.
- Arxiv:
- https://arxiv.org/abs/1603.08715
- HAL:
- https://hal.archives-ouvertes.fr/hal-01435034v1
- DOI:
- 10.1007/978-3-319-39300-1_5
- Abstract:
- We consider Turing machines as actions over configurations in $\Sigma^{\mathbb{Z}^d}$ which only change them locally around a marked position that can move and carry a particular state. In this setting we study the monoid of Turing machines and the group of reversible Turing machines. We also study two natural subgroups, namely the group of finite-state automata, which generalizes the topological full groups studied in the theory of orbit-equivalence, and the group of oblivious Turing machines whose movement is independent of tape contents, which generalizes lamplighter groups and has connections to the study of universal reversible logical gates. Our main results are that the group of Turing machines in one dimension is neither amenable nor residually finite, but is locally embeddable in finite groups, and that the torsion problem is decidable for finite-state automata in dimension one, but not in dimension two.
- Bibtex:
- @InProceedings{Barbieri_Kari_Salo_2016,

author="Barbieri, Sebasti{\'a}n and Kari, Jarkko and Salo, Ville",

editor="Cook, Matthew and Neary, Turlough",

title="The Group of Reversible Turing Machines",

booktitle="Cellular Automata and Discrete Complex Systems",

year="2016",

publisher="Springer International Publishing",

address="Cham",

pages="49--62",

isbn="978-3-319-39300-1"

}

- Book chapters
**[2018] About the Domino Problem for Subshifts on Groups.**

with Nathalie Aubrun and Emmanuel Jeandel.*In Sequences, Groups, and Number Theory.*

- File:
- Send me an e-mail (or use sci-hub).
- Book:
- Sequences, Groups, and Number Theory, 2018.
- HAL:
- https://hal.archives-ouvertes.fr/hal-01989760v1
- DOI:
- 10.1007/978-3-319-69152-7_9
- Abstract:
- From a classical point of view, the domino problem is the question of the existence of an algorithm which can decide whether a finite set of square tiles with colored edges can tile the plane, subject to the restriction that adjacent tiles share the same color along their adjacent edges. This question has already been settled in the negative by Berger in 1966, however, these tilings can be reinterpreted in dynamical terms using the formalism of subshifts of finite type, and hence the same question can be formulated for arbitrary finitely generated groups. In this chapter we present the state of the art concerning the domino problem in this extended framework. We also discuss different notions of effectiveness in subshifts defined over groups, that is, the ways in which these dynamical objects can be described through Turing machines.
- Bibtex:
- @incollection{Aubrun_Barbieri_Jeandel_2018,

address = {Cham},

series = {Trends in {Mathematics}},

title = {About the {Domino} {Problem} for {Subshifts} on {Groups}},

isbn = {978-3-319-69152-7},

language = {en},

urldate = {2019-05-25},

booktitle = {Sequences, {Groups}, and {Number} {Theory}},

publisher = {Springer International Publishing},

author = {Aubrun, Nathalie and Barbieri, Sebasti{\'{a}}n and Jeandel, Emmanuel},

editor = {Berth{\'{e}}, Val{\'{e}}rie and Rigo, Michel},

year = {2018},

doi = {10.1007/978-3-319-69152-7_9},

pages = {331--389},

}

- Thesis and Memoires
**[2017] Shift spaces on groups: computability and dynamics**, Thesis.

- File:
- [Tesis.pdf]
- Directors:
- Nathalie Aubrun and Stéphan Thomassé.
- Defense:
- ENS de Lyon, June 2017
- Report:
- [report_thesis.pdf]
- THESES:
- http://www.theses.fr/2017LYSEN021
- HAL:
- https://tel.archives-ouvertes.fr/tel-01563302/
- Abstract:
- Shift spaces are sets of colorings of a group which avoid a set of forbidden patterns and are endowed with a shift action. These spaces appear naturally as discrete versions of dynamical systems: they are obtained by partitioning the phase space and mapping each element into the sequence of partitions visited by its orbit. Several breakthroughs in this domain have pointed out the intricate relationship between dynamics of shift spaces and their computability properties. One remarkable example is the classification of the entropies of multidimensional subshifts of finite type as the set of right recursively enumerable numbers. This work explores shift spaces with a dual approach: on the one hand we are interested in their dynamical properties and on the other hand we study these objects as computational models. Four salient results have been obtained as a result of this approach: (1) a combinatorial condition ensuring non-emptiness of subshifts on arbitrary countable groups; (2) a simulation theorem which realizes effective actions of finitely generated groups as factors of a subaction of a subshift of finite type; (3) a characterization of effectiveness with oracles using generalized Turing machines and (4) the undecidability of the torsion problem for two group invariants of shift spaces. As byproducts of these results we obtain a simple proof of the existence of strongly aperiodic subshifts in countable groups. Furthermore, we realize them as subshifts of finite type in the case of a semidirect product of a $d$-dimensional integer lattice with a finitely generated group with decidable word problem whenever $d>1$.
- Bibtex:
- @phdthesis{barbierilemp:tel-01563302,

TITLE = {{Shift spaces on groups : computability and dynamics}},

AUTHOR = {Barbieri, Sebasti{\'{a}}n},

URL = {https://tel.archives-ouvertes.fr/tel-01563302},

NUMBER = {2017LYSEN021},

SCHOOL = {{Universit{\'e} de Lyon}},

YEAR = {2017},

MONTH = Jun,

KEYWORDS = {Conjugacy invariants ; Group theory ; Simulation theorems ; Symbolic dynamics ; Dynamical systems ; Shift spaces ; Aperiodicity ; Computability ; Dynamique symbolique ; Syst{\`e}mes dynamiques ; Sous-d{\'e}calages ; Aperiodicit{\'e} ; Calculabilit{\'e} ; Th{\'e}or{\`e}mes de simulation ; Th{\'e}orie des groupes ; Invariants de conjugaison},

TYPE = {Theses},

HAL_ID = {tel-01563302},

HAL_VERSION = {v1},

}**[2014] Tilings on different structures: exploration towards two problems**, Mémoire M2.

- File:
- [MemoriaFrancia.pdf]
- Director:
- Nathalie Aubrun.
- Defense:
- ENS de Lyon, June 2014
- Abstract:
- We study the problem of tiling structures which are different from the usual group $\mathbb{Z}^d$. In the first part we show a class of finitely generated groups for which the $G$-subshift $X_{\leq 1}$, which consists on the functions from $G$ to $\{0,1\}$ so that at most one $g \in G$ can map to 1, is not of sofic type. In the second part we study tilings over structures which are not groups and are generated by a special type of substitution. We define the emptiness problem and the possibility to simulate more complex substitutions in these structures and we show results in that direction for two specific examples. We end that section by constructing a partial order which under the assumption of a property preserves decidability of the emptiness problem monotonically.
**[2014] Subshifts generados por sustituciones multidimensionales**,*Memoria ingeniería Universidad de Chile*.

## Work in progress * ( last updated Nov 2019 ) *

- A long version of the Automata paper "The group of reversible Turing machines". With Jarkko Kari and Ville Salo.
- A generalization of the simulation theorem of Aubrun and Sablik to products of arbitrary finitely generated groups. This is work in progress with Mathieu Sablik.
- Extending the results about the domino problem for surface groups to word-hyperbolic groups and other classes. This is work in progress with Nathalie Aubrun and Laurent Bartholdi.
- Studying the symbolic dynamical systems that can arise from polygonal partitions of the torus and the $\mathbb{Z}^2$-translation action. This is joint work with Sébastien Labbé.
- Classifying the class of asymptotic pairs in symbolic systems which are statistically indistinguishable from the point of view of word combinatorics. This is work in progress with Sébastien Labbé and Štěpán Starosta.