arXiv subjects

Corentin Bodart

Publications and source records attributed to Corentin Bodart.

14 recordsLinked to original sources

arXiv2026

A graph-theoretical characterisation of subgroups of Thompson's group $V$

We prove a graph-theoretical characterisation of finitely generated subgroups of Thompson's group $V$: a finitely generated group embeds in $V$ if and only if it admits a faithful context-free action, or equivalently if it belongs to the class CF-TR of transition groups of context-free graphs recently introduced by Matucci and the three last authors. Using this characterisation, we prove results in different directions: - All known examples of groups with co-context-free Word Problem do embed in $V$, providing evidence towards Lehnert's conjecture. - Each finitely generated subgroup of $V$ is either virtually abelian, or contains a free non-abelian semigroup. It follows that groups of intermediate growth do not embed in Thompson's $V$. We further study the relation between transition groups defined by graphs that are limits or covers of each others, and prove properties of transition groups of context-free graphs of polynomial growth. Finally, we prove that the Basilica and Hano\"i Towers groups do not embed in $V$. This uses the geometry of Schreier graphs of the natural actions of these groups and of Thompson's $V$.

math.GR↗

arXiv2026

Period growth and co-context-free groups

We study period growth in co-context-free groups, giving general results and looking at specific examples such as Thompson groups $T$ and $V$ and the Houghton groups $H_m$. Along the way, we give a refined upper bound on the word metric in Thompson $V$, as well as efficient algorithms to determine if elements of $V$ are torsion, and compute their order. We also adapt our algorithm to compute the rotation number of elements of $T$ and answer a question of D. Calegari.

math.GR↗

arXiv2025

Groups with a finite Busemann boundary are virtually cyclic

This note is a continuation of the study of the relationship between the geometry of Cayley graphs and the size of its metric-functional boundary. We show that, if there exists a Cayley graph with finitely many Busemann points, then the underlying group is virtually cyclic. Together with previous works, this completes the full characterization of groups with finite metric-functional boundaries. The main new notion introduced is that of annihilators.

math.GR↗

arXiv2025

Twisted commutativity and conjugacy ratio in groups

In this paper we introduce and study the degree of twisted commutativity and the twisted conjugacy ratio of a finitely generated group $G$. The degree of twisted commutativity $\mathrm{tdc}_X(\varphi, G)$ generalises the degree of commutativity of $G$, by measuring the density of pairs of elements with trivial twisted commutators in the ball of radius $n$ of $G$, as $n \rightarrow \infty$, where the twisting is done with respect to an endomorphism $\varphi$ of $G$. We compute $\mathrm{tdc}_X(\varphi, G)$ for several classes of groups, including virtually abelian groups, groups of subexponential growth, and free groups. We then study the twisted conjugacy ratio $\mathrm{tcr}_{X}(\varphi, G)$, which is the limit at infinity of the quotient of the twisted conjugacy and standard growth functions. We compute $\mathrm{tcr}_{X}(\varphi, G)$ for virtually abelian groups, and give examples of groups of exponential growth such that $\mathrm{tcr}_{X}(\varphi, G) = 0$.

math.GR↗

arXiv2025

On finite extensions of lamplighter groups

We study a family of groups consisting of the simplest extensions of lamplighter groups. We use these groups to answer multiple open questions in combinatorial group theory, providing groups that exhibit various combinations of properties: 1) Decidable Subgroup Membership and undecidable Uniform Subgroup Membership Problem, 2) Rational volume growth series and undecidable Word Problem and 3) Recursive (even context-free) language of conjugacy geodesics, decidable Word Problem, and undecidable Conjugacy Problem. We also consider the co-Word Problem, residual finiteness and the Isomorphism Problem within this class.

math.GR↗

arXiv2024

Ordering groups and the Identity Problem

In this paper, the Identity Problem for certain groups, which asks if the subsemigroup generated by a given finite set of elements contains the identity element, is related to problems regarding ordered groups. Notably, the Identity Problem for a torsion-free nilpotent group corresponds to the problem asking if a given finite set of elements extends to the positive cone of a left-order on the group, and thereby also to the Word Problem for a related lattice-ordered group. A new (independent) proof is given showing that the Identity and Subgroup Problems are decidable for every finitely presented nilpotent group, establishing also the decidability of the Word Problem for a family of lattice-ordered groups. A related problem, the Fixed-Target Submonoid Membership Problem, is shown to be undecidable in nilpotent groups. Decidability of the Normal Identity Problem (with `subsemigroup' replaced by `normal subsemigroup') for free nilpotent groups is established using the (known) decidability of the Word Problem for certain lattice-ordered groups. Connections between orderability and the Identity Problem for a class of torsion-free metabelian groups are also explored.

math.GR↗

arXiv2024

A virtually nilpotent group whose Green series is not D-finite

We provide the first example of virtually nilpotent group, with a specific generating set, for which the Green series (sometimes called cogrowth series) is not $D$-finite. The proof relies on an arithmetical miracle, and the study of the subword complexity of a multiplicative sequence coming out of it.

math.GR↗

arXiv2024

Horofunctions on the Heisenberg and Cartan groups

We study the horofunction boundary of finitely generated nilpotent groups, and the natural group action on it. More specifically, we prove the followings results: For discrete Heisenberg groups, we classify the orbits of Busemann points. As a byproduct, we observe that the set of orbits is finite and the set of Busemann points is countable. Furthermore, using the approximation with Lie groups, we observe that the entire horoboundary is uncountable. For the discrete Cartan group, we exhibit an continuum of Busemann points, disproving a conjecture of Tointon and Yadin. As a byproduct, we prove that the group acts non-trivially on its reduced horoboundary, disproving a conjecture of Bader and Finkelshtein.

math.GR↗

arXiv2024

Finitely presented groups with transcendental spectral radius

We provide examples of groups with transcendental spectral radius: We first construct finitely presented examples, using links between decidability of the Word Problem and semi-computability of the spectral radius. This argument extends to the exponential growth rate and the asymptotic entropy. We also construct a finitely generated example with decidable Word Problem, using classical small-cancellation theory. Along the way, we prove that $C'(1/6)$ groups satisfy the Rapid Decay property, and deduce some properties on their spectral radii of independent interest.

math.GR↗

arXiv2024

The Identity Problem in virtually solvable matrix groups over algebraic numbers

The Tits alternative states that a finitely generated matrix group either contains a nonabelian free subgroup $F_2$, or it is virtually solvable. This paper considers two decision problems in virtually solvable matrix groups: the Identity Problem (does a given finitely generated subsemigroup contain the identity matrix?), and the Group Problem (is a given finitely generated subsemigroup a group?). We show that both problems are decidable in virtually solvable matrix groups over the field of algebraic numbers $\overline{\mathbb{Q}}$. Our proof also extends the decidability result for nilpotent groups by Bodart, Ciobanu, Metcalfe and Shaffrir, and the decidability result for metabelian groups by Dong (STOC'24). Since the Identity Problem and the Group Problem are known to be undecidable in matrix groups containing $F_2 \times F_2$, our result significantly reduces the decidability gap for both decision problems.

math.GR↗

arXiv2024

Membership problems in nilpotent groups

We study both the Submonoid Membership problem and the Rational Subset Membership problem in finitely generated nilpotent groups. We give two reductions with important applications. First, Submonoid Membership in any nilpotent group can be reduced to Rational Subset Membership in smaller groups. As a corollary, we prove the existence of a group with decidable Submonoid Membership and undecidable Rational Subset Membership, confirming a conjecture of Lohrey and Steinberg. Second, the Rational Subset Membership problem in $H_3(\mathbb Z)$ can be reduced to the Knapsack problem in the same group, and is therefore decidable. Combining both results, we deduce that the filiform $3$-step nilpotent group has decidable Submonoid Membership.

math.GR↗

arXiv2023

Intermediate geodesic growth in virtually nilpotent groups

We give a criterion on pairs $(G,S)$ - where $G$ is a virtually $s$-step nilpotent group and $S$ is a finite generating set - saying whether the geodesic growth is exponential or strictly sub-exponential. Whenever $s=1,2$, this goes further and we prove the geodesic growth is either exponential or polynomial. For $s\ge 3$ however, intermediate growth is possible. We provide an example of virtually $3$-step nilpotent group for which $\gamma_{\mathrm{geod}}(n) \asymp \exp\!\big(n^{3/5}\cdot \log(n)\big)$. This is the first known example of group with intermediate geodesic growth. Along the way, we prove results on the geometry of virtually nilpotent groups, including asymptotics with error terms for their volume growth.

math.GR↗

arXiv2022

Dead ends and rationality of complete growth series

The complete growth series of a finitely generated group is given by $\sum_{n\ge 0} A_ns^n$, where $A_n$ is the sum of elements of length $n$ in the group semiring. We study the $\mathbb NG$-rationality and $\mathbb NG$-algebraicity of such series. We show that having dead ends of arbitrarily large depths is an obstruction to $\mathbb NG$-rationality. In the case of the $3$-dimensional Heisenberg group $H_3(\mathbb Z)$, we prove that the complete series is not $\mathbb NG$-algebraic for any generating set. Dead ends are also used to show that complete growth series of higher Heisenberg groups are not $\mathbb NG$-rational for specific generating sets. Using a more general version of this obstruction, we prove that complete growth series of some lamplighter groups are not $\mathbb NG$-rational either.

math.GR↗

arXiv2022

Rational cross-sections, bounded generation and orders on groups

We provide new examples of groups without rational cross-sections (also called regular normal forms), using connections with bounded generation and rational orders on groups. Specifically, our examples are extensions of infinite torsion groups, groups of Grigorchuk type, wreath products similar to $C_2\wr(C_2\wr \mathbb Z)$ and $\mathbb Z\wr F_2$, a group of permutations of $\mathbb Z$, and a finitely presented HNN extension of the first Grigorchuk group. This last group is the first example of finitely presented group with solvable word problem and without rational cross-sections. It is also not autostackable, and has no left-regular complete rewriting system.

math.GR↗