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Dominik Francoeur

Publications and source records attributed to Dominik Francoeur.

At least 19 recordsLinked to original sources
arXiv2026

Stallings foldings for rational subsets of automatic groups

Let $G$ be an automatic group with associated regular language $L$. We describe a procedure for constructing an automaton which recognises elements of a given submonoid or rational subset $K$ of $G$. This builds on work of Kharlampovich, Miasnikov and Weil, on the case where $K$ is a subgroup of $G$. Our construction succeeds, after sufficiently many iterations, whenever $K$ satisfies a certain convexity property, which we call $L$-proximity. We show how to test whether the construction is complete in the case that $K$ is a submonoid; we have no such test for the general case of a rational subset $K$. We focus particularly on the case of a surface group $G$ of genus $g>1$, where $L$ is the language of geodesic words in the standard generators. We use small cancellation theory to obtain a method for constructing $L$-recognisable submonoids of $G$.

math.GR↗
arXiv2025

On bireversible automata and commensurators of groups in automorphisms of their Cayley graphs

If $G$ is a finitely generated group and $X$ is a Cayley graph of $G$, denote by $\mathcal{C}_1^X(G)$ the subgroup of all automorphisms of $X$ commensurating $G$ and fixing the vertex corresponding to the identity. Building on the work of Macedońska, Nekrashevych and Sushchansky, we observe that $\mathcal{C}_1^X(G)$ can be expressed as a directed union of groups generated by bireversible automata. We use this to to show that every cyclic subgroup of $\mathcal{C}_1^X(G)$ is undistorted and to obtain a necessary condition on $G$ for $\mathcal{C}_1^X(G)$ not to be locally finite. As a consequence, we prove that several families of groups cannot be generated by bireversible automata and show that the set of groups generated by bireversible automata is strictly contained in the set of groups generated by invertible and reversible automata.

math.GR↗
arXiv2025

On closure operations in the space of subgroups and applications

We establish some interactions between uniformly recurrent subgroups (URSs) of a group $G$ and cosets topologies $τ_\mathcal{N}$ on $G$ associated to a family $\mathcal{N}$ of normal subgroups of $G$. We show that when $\mathcal{N}$ consists of finite index subgroups of $G$, there is a natural closure operation $\mathcal{H} \mapsto \mathrm{cl}_\mathcal{N}(\mathcal{H})$ that associates to a URS $\mathcal{H}$ another URS $\mathrm{cl}_\mathcal{N}(\mathcal{H})$, called the $τ_\mathcal{N}$-closure of $\mathcal{H}$. We give a characterization of the URSs $\mathcal{H}$ that are $τ_\mathcal{N}$-closed in terms of stabilizer URSs. This has consequences on arbitrary URSs when $G$ belongs to the class of groups for which every faithful minimal profinite action is topologically free. We also consider the largest amenable URS $\mathcal{A}_G$, and prove that for certain coset topologies on $G$, almost all subgroups $H \in \mathcal{A}_G$ have the same closure. For groups in which amenability is detected by a set of laws (a property that is variant of the Tits alternative), we deduce a criterion for $\mathcal{A}_G$ to be a singleton based on residual properties of $G$.

math.GR↗
arXiv2025

Subgroup induction property for branch groups

The subgroup induction property is a property of self-similar groups acting on rooted trees introduced by Grigorchuk and Wilson in 2003 that appears to have strong implications on the structure of the groups possessing it. It was for example used in the proof that the first Grigorchuk group as well as the Gupta-Sidki 3-group are subgroup separable (locally extended residually finite) or to describe their finitely generated subgroups as well as their weakly maximal subgroups. However, until now, there were only two known examples of groups with this property, namely the first Grigorchuk group and the Gupta-Sidki 3-group. The aim of this article is twofold. First, we investigate various consequences of the subgroup induction property for branch groups, a particularly interesting class of self-similar groups. Notably, we show that finitely generated branch groups with the subgroup induction property must be torsion, just infinite and subgroup separable, and we establish conditions under which all their maximal subgroups are of finite index and all their weakly maximal subgroups are closed in the profinite topology. Then, we show that every torsion GGS group has the subgroup induction property, hence providing the first infinite family of examples of groups with this property.

math.GR↗
arXiv2023

Intersection-saturated groups without free subgroups

A group $G$ is said to be intersection-saturated if for every strictly positive integer $n$ and every map $c\colon \mathcal{P}(\{1,\dots, n\})\setminus \emptyset \rightarrow \{0,1\}$, one can find subgroups $H_1,\dots, H_n\leq G$ such that for every non-empty subset $I\subseteq \{1,\dots, n\}$, the intersection $\bigcap_{i\in I}H_i$ is finitely generated if and only if $c(I)=0$. We obtain a new criterion for a group to be intersection-saturated based on the existence of arbitrarily high direct powers of a subgroup admitting an automorphism with a non-finitely generated set of fixed points. We use this criterion to find new examples of intersection-saturated groups, including Thompson's groups and the Grigorchuk group. In particular, this proves the existence of finitely presented intersection-saturated groups without non-abelian free subgroups, thus answering a question of Delgado, Roy and Ventura.

math.GR↗
arXiv2021

On quasi-2-transitive actions of branch groups

An action of a group $G$ on a set $X$ is said to be quasi-n-transitive if the diagonal action of $G$ on $X^n$ has only finitely many orbits. We show that branch groups, a special class of groups of automorphisms of rooted trees, cannot act quasi-2-transitively on infinite sets.

math.GR↗
arXiv2021

Maximal subgroups of non-torsion Grigorchuk-Gupta-Sidki groups

A Grigorchuk-Gupta-Sidki (GGS-)group is a subgroup of the automorphism group of the $p$-adic tree for an odd prime $p$, generated by one rooted automorphism and one directed automorphism. Pervova proved that all torsion GGS-groups do not have maximal subgroups of infinite index. Here we extend the result to non-torsion GGS-groups, which include the weakly regular branch, but not branch, GGS-group.

math.GR↗
arXiv2021

Commensurated subgroups and micro-supported actions

Let $Γ$ be a finitely generated group and $X$ be a minimal compact $Γ$-space. We assume that the $Γ$-action is micro-supported, i.e. for every non-empty open subset $U \subseteq X$, there is an element of $Γ$ acting non-trivially on $U$ and trivially on the complement $X \setminus U$. We show that, under suitable assumptions, the existence of certain commensurated subgroups in $Γ$ yields strong restrictions on the dynamics of the $Γ$-action: the space $X$ has compressible open subsets, and it is an almost $Γ$-boundary. Those properties yield in turn restrictions on the structure of $Γ$: $Γ$ is neither amenable nor residually finite. Among the applications, we show that the (alternating subgroup of the) topological full group associated to a minimal and expansive Cantor action of a finitely generated amenable group has no commensurated subgroups other than the trivial ones. Similarly, every commensurated subgroup of a finitely generated branch group is commensurate to a normal subgroup; the latter assertion relies on an appendix by Dominik Francoeur, and generalizes a result of Phillip Wesolek on finitely generated just-infinite branch groups. Other applications concern discrete groups acting on the circle, and the centralizer lattice of non-discrete totally disconnected locally compact (tdlc) groups. Our results rely, in an essential way, on recent results on the structure of tdlc groups, on the dynamics of their micro-supported actions, and on the notion of uniformly recurrent subgroups.

math.GR↗
arXiv2020

Infinite Automaton Semigroups and Groups Have Infinite Orbits

We show that an automaton group or semigroup is infinite if and only if it admits an $ω$-word (i. e. a right-infinite word) with an infinite orbit, which solves an open problem communicated to us by Ievgen V. Bondarenko. In fact, we prove a generalization of this result, which can be applied to show that finitely generated subgroups and subsemigroups as well as principal left ideals of automaton semigroups are infinite if and only if there is an $ω$ -word with an infinite orbit under their action. The proof also shows some interesting connections between the automaton semigroup and its dual. Finally, our result is interesting from an algorithmic perspective as it allows for a reformulation of the finiteness problem for automaton groups and semigroups.

cs.FL↗
arXiv2020

On the Orbits of Automaton Semigroups and Groups

We investigate the orbits of automaton semigroups and groups to obtain algorithmic and structural results, both for general automata but also for some special subclasses. First, we show that a more general version of the finiteness problem for automaton groups is undecidable. This problem is equivalent to the finiteness problem for left principal ideals in automaton semigroups generated by complete and reversible automata. Then, we look at $ω$-word (i.e. right infinite words) with a finite orbit. We show that every automaton yielding an $ω$-word with a finite orbit already yields an ultimately periodic one, which is not periodic in general, however. On the algorithmic side, we observe that it is not possible to decide whether a given periodic $ω$-word has an infinite orbit and that we cannot check whether a given reversible and complete automaton admits an $ω$-word with a finite orbit, a reciprocal problem to the finiteness problem for automaton semigroups in the reversible case. Finally, we look at automaton groups generated by reversible but not bi-reversible automata and show that many words have infinite orbits under the action of such automata.

cs.FL↗
arXiv2020

On the stabilisers of points in groups with micro-supported actions

Given a group $G$ of homeomorphism of a first-countable Hausdorff space $\mathcal{X}$, we prove that if the action of $G$ on $\mathcal{X}$ is minimal and has rigid stabilisers that act locally minimally, then the neighbourhood stabilisers of any two points in $\mathcal{X}$ are conjugated by a homeomorphism of $\mathcal{X}$. This allows us to study stabilisers of points in many classes of groups, such as topological full groups of Cantor minimal systems, Thompson groups, branch groups, and groups acting on trees with almost prescribed local actions.

math.GR↗
arXiv2020

On maximal subgroups of infinite index in branch and weakly branch groups

We generalise a technical tool, originally developed by Pervova for the study of maximal subgroups in Grigorchuk and GGS groups, to all weakly branch groups satisfying a natural condition, and in particular to all branch groups. We then use this tool to prove that every maximal subgroup of infinite index of a branch group is also a branch group. As a further application of this result, we show that every maximal subgroup of the Basilica group is of finite index.

math.GR↗
arXiv2019

Einstein tori and crooked surfaces

In hyperbolic space, the angle of intersection and distance classify pairs of totally geodesic hyperplanes. A similar algebraic invariant classifies pairs of hyperplanes in the Einstein universe. In dimension 3, symplectic splittings of a 4-dimensional real symplectic vector space model Einstein hyperplanes and the invariant is a determinant. The classification contributes to a complete disjointness criterion for crooked surfaces in the 3-dimensional Einstein universe.

math.DG↗
arXiv2018

Maximal subgroups of groups of intermediate growth

Finding the number of maximal subgroups of infinite index of a finitely generated group is a natural problem that has been solved for several classes of `geometric' groups (linear groups, hyperbolic groups, mapping class groups, etc). Here we provide a solution for a family of groups with a different geometric origin: groups of intermediate growth that act on rooted binary trees. In particular, we show that the non-torsion iterated monodromy groups of the tent map (a special case of some groups first introduced by {Š}uni{ć} in \cite{Sunic} as `siblings of the Grigorchuk group') have exactly countably many maximal subgroups of infinite index, and describe them up to conjugacy. This is in contrast to the torsion case (e.g. Grigorchuk group) where there are no maximal subgroups of infinite index. It is also in contrast to the above-mentioned geometric groups, where there are either none or uncountably many such subgroups. Along the way we show that all the groups defined by {Š}uni{ć} have the congruence subgroup property and are just infinite.

math.GR↗