arXiv subjects

Francesco Fournier-Facio

Publications and source records attributed to Francesco Fournier-Facio.

At least 37 records · Page 2Linked to original sources

arXiv2026

Finiteness properties of stabilisers of oligomorphic actions

An action of a group on a set is oligomorphic if it has finitely many orbits of $n$-element subsets for all $n$. We prove that for a large class of groups (including all groups of finite virtual cohomological dimension and all countable linear groups), for any oligomorphic action of such a group on an infinite set there exists a finite subset whose stabiliser is not of type $\mathrm{FP}_\infty$. This leads to obstructions on finiteness properties for permutational wreath products and twisted Brin-Thompson groups. We also prove a version for actions on flag complexes, and discuss connections to the Boone-Higman conjecture. In the appendix, we improve on the criterion of Bartholdi-Cornulier-Kochloukova for finiteness properties of wreath products, and the criterion of Kropholler-Martino for finiteness properties of graph-wreath products.

math.GR↗

arXiv2026

Property $R_{\infty}$ for groups with infinitely many ends

We show that an accessible group with infinitely many ends has property $R_{\infty}$. That is, it has infinitely many twisted conjugacy classes for any twisting automorphism. We deduce that having property $R_{\infty}$ is undecidable amongst finitely presented groups. We also show that the same is true for a wide class of relatively hyperbolic groups, filling in some of the gaps in the literature. Specifically, we show that a non-elementary, finitely presented relatively hyperbolic group with finitely generated peripheral subgroups which are not themselves relatively hyperbolic, has property $R_{\infty}$.

math.GR↗

arXiv2025

Stable commutator length on free $\mathbb{Q}$-groups

We study stable commutator length on free $\mathbb{Q}$-groups. We prove that every non-identity element has positive stable commutator length, and that the corresponding free group embeds isometrically. We deduce that a non-abelian free $\mathbb{Q}$-group has an infinite-dimensional space of homogeneous quasimorphisms modulo homomorphisms, answering a question of Casals-Ruiz, Garreta, and de la Nuez Gonz{á}lez. We conjecture that stable commutator length is rational on free $\mathbb{Q}$-groups. This is connected to the long-standing problem of rationality on surface groups: indeed, we show that free $\mathbb{Q}$-groups contain isometrically embedded copies of non-orientable surface groups.

math.GR↗

arXiv2025

First-order theory of torsion-free Tarski monsters

We develop methods to control the first-order theory of groups arising as certain direct limits of torsion-free hyperbolic groups, answering several questions in the literature. We construct simple torsion-free Tarski monsters $Γ$ (non-abelian groups whose non-trivial, proper subgroups are infinite cyclic) that are $\exists \forall \exists$-elementarily embedded into $Γ\ast \mathbf{Z}$. In particular, such $Γ$ have the same two-quantifier theory as $Γ\ast \mathbf{Z}$, and hence the same positive theory as a non-abelian free group. All previously known examples of groups with the same positive theory as the free group admit a non-elementary action on a hyperbolic space, while our examples cannot act on a hyperbolic space with a loxodromic element. Along the way, we solve the one-quantifier Knight conjecture for random quotients of arbitrary torsion-free, non-elementary, hyperbolic groups in the few-relator model.

math.GR↗

arXiv2025

An algebraic criterion for the vanishing of bounded cohomology

We prove the vanishing of bounded cohomology with separable dual coefficients for many groups of interest in geometry, dynamics, and algebra. These include compactly supported structure-preserving diffeomorphism groups of certain manifolds; the group of interval exchange transformations of the half line; piecewise linear and piecewise projective groups of the line, giving strong answers to questions of Calegari and Navas; direct limit linear groups of relevance in algebraic $K$-theory, thereby answering a question by Kastenholz and Sroka and a question of two of the authors and L{ö}h; and certain subgroups of big mapping class groups, such as the stable braid group and the stable mapping class group, proving a conjecture of Bowden. Moreover, we prove that in the recently introduced framework of enumerated groups, the generic group has vanishing bounded cohomology with separable dual coefficients. At the heart of our approach is an elementary algebraic criterion called the commuting cyclic conjugates condition that is readily verifiable for the aforementioned large classes of groups.

math.GR↗

arXiv2025

Dimensions of finitely generated simple groups and their subgroups

We construct finitely generated simple torsion-free groups with strong homological control. Our main result is that every subset of $\mathbb{N} \cup \{\infty\}$, with some obvious exceptions, can be realized as the set of dimensions of subgroups of a finitely generated simple torsion-free group. This is new even for basic cases such as $\{ 0, 1, 3 \}$ and $\{ 0, 1, \infty \}$, even without simplicity or finite generation, and answers a question of Talelli and disproves a conjecture of Petrosyan. Moreover, we prove that every countable group of dimension at least $2$ embeds into a finitely generated simple group of the same dimension. These are the first examples of finitely generated simple groups with dimension other than $2$ or $\infty$. As another application, we exhibit the first examples of torsion-free groups with the fixed point property for actions on finite-dimensional contractible CW-complexes, and construct torsion-free groups in all countable levels of Kropholler's hierarchy, answering a question of Januszkiewicz, Kropholler and Leary. Our method combines small cancellation theory with group theoretic Dehn filling, and allows to do several other exotic constructions with control on the dimension. Along the way we construct the first uncountable family of pairwise non-measure equivalent finitely generated torsion-free groups.

math.GR↗

arXiv2025

Stable reflection length in Coxeter groups

We introduce stable reflection length in Coxeter groups, as a way to study the asymptotic behaviour of reflection length. This creates connections to other well-studied stable length functions in groups, namely stable commutator length and stable torsion length. As an application, we give a complete characterisation of elements whose reflection length is unbounded on powers.

math.GR↗

arXiv2025

A note on separability in outer automorphism groups

We give a criterion for separability of subgroups of certain outer automorphism groups. This answers questions of Hagen and Sisto, by strengthening and generalizing a result of theirs on mapping class groups.

math.GR↗

arXiv2024

Local Hilbert--Schmidt stability

We introduce a notion of local Hilbert--Schmidt stability, motivated by the recent definition by Bradford of local permutation stability, and give examples of (non-residually finite) groups that are locally Hilbert--Schmidt stable but not Hilbert--Schmidt stable. For amenable groups, we provide a criterion for local Hilbert--Schmidt stability in terms of group characters, by analogy with the character criterion of Hadwin and Shulman for Hilbert--Schmidt stable amenable groups. Furthermore, we study the (very) flexible analogues of local Hilbert--Schmidt stability, and we prove several results analogous to the classical setting. Finally, we prove that infinite sofic, respectively hyperlinear, property (T) groups are never locally permutation stable, respectively locally Hilbert--Schmidt stable. This strengthens the result of Becker and Lubotzky for classical stability, and answers a question of Lubotzky.

math.GR↗

arXiv2024

Displacement techniques in bounded cohomology

Several algebraic criteria, reflecting displacement properties of transformation groups, have been used in the past years to prove vanishing of bounded cohomology and stable commutator length. Recently, the authors introduced the property of commuting cyclic conjugates, a new displacement technique that is widely applicable and provides vanishing of the bounded cohomology in all positive degrees and all dual separable coefficients. In this note we consider the most recent along with the by now classical displacement techniques and we study implications among them as well as counterexamples.

math.GR↗

arXiv2024

Hopfian wreath products and the stable finiteness conjecture

We study the Hopf property for wreath products of finitely generated groups, focusing on the case of an abelian base group. Our main result establishes a strong connection between this problem and Kaplansky's stable finiteness conjecture. Namely, the latter holds true if and only if for every finitely generated abelian group $A$ and every finitely generated Hopfian group $Γ$ the wreath product $A \wr Γ$ is Hopfian. In fact, we characterize precisely when $A \wr Γ$ is Hopfian, in terms of the existence of one-sided units in certain matrix algebras over $\mathbb{F}_p[Γ]$, for every prime $p$ occurring as the order of some element in $A$. A tool in our arguments is the fact that fields of positive characteristic locally embed into matrix algebras over $\mathbb{F}_p$ thus reducing the stable finiteness conjecture to the case of $\mathbb{F}_p$. A further application of this result shows that the validity of Kaplansky's stable finiteness conjecture is equivalent to a version of Gottschalk's surjunctivity conjecture for additive cellular automata.

math.GR↗

arXiv2023

Median quasimorphisms on CAT(0) cube complexes and their cup products

Cup products provide a natural approach to access higher bounded cohomology groups. We extend vanishing results on cup products of Brooks quasimorphisms of free groups to cup products of median quasimorphisms, i.e., Brooks-type quasimorphisms of group actions on CAT(0) cube complexes. In particular, we obtain such vanishing results for groups acting on trees and for right-angled Artin groups. Moreover, we outline potential applications of vanishing results for cup products in bounded cohomology.

math.GR↗

arXiv2023

GAGTA 2023 Problem Session

The conference `Geometric and Asymptotic Group Theory with Applications (GAGTA) 2023: Groups and Dynamics' took place at the Erwin Schrödinger Institute on July 17-21. These are the problems that were proposed during the Problem Session: Residual finiteness growth, Geometric v. random walk boundaries, Stability, Cogrowth, Embedding left orderable groups, Schreier growth gap, $\ell^p$ models for hyperbolic groups, Characterizing hyperbolic groups

math.GR↗

arXiv2023

Property (NL) for group actions on hyperbolic spaces

We introduce Property (NL), which indicates that a group does not admit any (isometric) action on a hyperbolic space with loxodromic elements. In other words, such a group $G$ can only admit elliptic or horocyclic hyperbolic actions, and consequently its poset of hyperbolic structures $\mathcal{H}(G)$ is trivial. It turns out that many groups satisfy this property; and we initiate the formal study of this phenomenon. Of particular importance is the proof of a dynamical criterion in this paper that ensures that groups with rich actions on compact Hausdorff spaces have Property (NL). These include many Thompson-like groups, such as $V, T$ and even twisted Brin--Thompson groups, which implies that every finitely generated group quasi-isometrically embeds into a finitely generated simple group with Property (NL). We also study the stability of the property under group operations and explore connections to other fixed point properties. In the appendix (by Alessandro Sisto) we describe a construction of cobounded actions on hyperbolic spaces starting from non-cobounded ones that preserves various properties of the initial action.

math.GR↗

arXiv2023

Ulam stability of lamplighters and Thompson groups

We show that a large family of groups is uniformly stable relative to unitary groups equipped with submultiplicative norms, such as the operator, Frobenius, and Schatten $p$-norms. These include lamplighters $Γ\wr Λ$ where $Λ$ is infinite and amenable, as well as several groups of dynamical origin such as the classical Thompson groups $F, F', T$ and $V$. We prove this by means of vanishing results in asymptotic cohomology, a theory introduced by the second author, Glebsky, Lubotzky and Monod, which is suitable for studying uniform stability. Along the way, we prove some foundational results in asymptotic cohomology, and use them to prove some hereditary features of Ulam stability. We further discuss metric approximation properties of such groups, taking values in unitary or symmetric groups.

math.GR↗

arXiv2023

Finitely presented left orderable monsters

A left orderable monster is a finitely generated left orderable group all of whose fixpoint-free actions on the line are proximal: the action is semiconjugate to a minimal action so that for every bounded interval $I$ and open interval $J$, there is a group element that sends $I$ into $J$. In his 2018 ICM address, Navas asked about the existence of left orderable monsters. By now there are several examples, all of which are finitely generated but not finitely presentable. We provide the first examples of left orderable monsters that are finitely presentable, and even of type $F_\infty$. The construction itself is elementary, and these groups satisfy several additional properties separating them from the previous examples: they are not simple, they act minimally on the circle, and they have an infinite-dimensional space of homogeneous quasimorphisms. Our construction is flexible enough that it produces infinitely many isomorphism classes of finitely presented (and type $F_{\infty}$) left orderable monsters.

math.GR↗

arXiv2023

Second bounded cohomology of groups acting on $1$-manifolds and applications to spectrum problems

We prove a general criterion for the vanishing of second bounded cohomology (with trivial real coefficients) for groups that admit an action satisfying certain mild hypotheses. This leads to new computations of the second bounded cohomology for a large class of groups of homeomorphisms of $1$-manifolds, and a plethora of applications. First, we demonstrate that the finitely presented and nonamenable group $G_0$ constructed by the second author with Justin Moore satisfies that every subgroup has vanishing second bounded cohomology. This provides the first solution to a homological version of the von Neumann--Day Problem, posed by Calegari. Next, we develop a technical refinement of our criterion to demonstrate the existence of finitely generated non-indicable (even simple) left orderable groups with vanishing second bounded cohomology. This answers Question 8 from the 2018 ICM proceedings article of Navas. Then we provide the first examples of finitely presented groups whose spectrum of stable commutator length contains algebraic irrationals, answering a question of Calegari. Finally, we provide the first examples of manifolds whose simplicial volumes are algebraic and irrational, as further evidence towards a conjecture of Heuer and Löh.

math.GR↗

arXiv2023

Finitely generated simple left orderable groups with vanishing second bounded cohomology

We prove that the finitely generated simple left orderable groups constructed by the second author with Hyde have vanishing second bounded cohomology, both with trivial real and trivial integral coefficients. As a consequence, these are the first examples of finitely generated non-indicable left orderable groups with vanishing second bounded cohomology. This answers Question 8 from the 2018 ICM proceedings article of Andrés Navas.

math.GR↗