arXiv subjects

Francesco Fournier-Facio

Publications and source records attributed to Francesco Fournier-Facio.

At least 19 recordsLinked to original sources

arXiv2026

Higher Brin--Thompson groups are Kazhdan

We give an algebraic criterion for certain sequences $(G_n)_{n\in\mathbb{N}}$ of groups to have Kazhdan's property $(T)$ eventually. As a consequence, we prove that the Brin--Thompson groups $nV$ have Kazhdan's property $(T)$ for large $n$. The algebraic criterion provides a uniform proof that $nV$, $\mathrm{Aut}(F_n)$ and $\mathrm{EL}_n(R)$ for a finitely generated associative unital ring $R$ have property $(T)$ for large $n$. In an appendix, written by Francesco Fournier-Facio, it is shown that the group $nV$ for large $n$ is neither hyperlinear, in particular not sofic, nor $\mathrm{MF}$. Further, it is not Schatten $p$-approximable for any $1\le p<\infty$.

math.GR↗

arXiv2026

The bounded cohomology of transformation groups of Euclidean spaces and discs

We prove that the groups of orientation-preserving homeomorphisms and diffeomorphisms of $\mathbb{R}^n$ are boundedly acyclic, in all regularities. This is the first full computation of the bounded cohomology of a transformation group that is not compactly supported, and it implies that many characteristic classes of flat $\mathbb{R}^n$-bundles and $S^n$-bundles are unbounded. We obtain the same result for the group of homeomorphisms of the closed disc that restrict to the identity on the boundary, and for the homeomorphism group of the non-compact Cantor set. In the appendix, Alexander Kupers proves a controlled version of the annulus theorem which we use to study the bounded cohomology of the homeomorphism group of the discs.

math.GT↗

arXiv2026

Boone-Higman embeddings of $\mathrm{Aut}(F_n)$ and mapping class groups of punctured surfaces

We prove that the groups $\mathrm{Aut}(F_n)$ satisfy the Boone-Higman conjecture for all $n$, meaning each $\mathrm{Aut}(F_n)$ embeds in a finitely presented simple group. In fact, we prove that each $\mathrm{Aut}(F_n)$ satisfies the "permutational" Boone-Higman conjecture, which means the simple group in question can be taken to be a twisted Brin-Thompson group. A far-reaching consequence of our approach is that finitely presented twisted Brin-Thompson groups are universal among finitely presented simple groups that are highly transitive. This is evidence toward the Boone-Higman conjecture being equivalent to its permutational version. Proving the conjecture for $\mathrm{Aut}(F_n)$ also confirms the conjecture for all groups (virtually) embedding into some $\mathrm{Aut}(F_n)$, such as mapping class groups of non-closed surfaces, braid groups, loop braid groups, ribbon braid groups and certain Artin groups. This answers several questions of the first and fourth authors with Bleak and Matucci. Yet another consequence of our approach is that satisfying the permutational Boone-Higman conjecture is closed under free products and regular wreath products, which answers a question of the fourth author, confirms the conjecture for free solvable groups, and reduces the conjecture for finitely generated $3$-manifold groups to the case of non-Seifert fibered closed graph manifolds.

math.GR↗

arXiv2026

Infinite simple characteristic quotients

We construct continuum many infinite, simple, characteristic quotients of non-abelian free groups, answering a 1978 question of James Wiegold. The method is very flexible, allowing to impose certain properties on the quotients, to generalize the construction to large classes of groups with hyperbolic features, and to produce quasi-isometrically diverse examples.

math.GR↗

arXiv2026

A torsion-free non-sofic group

OpenAI announced the existence of a non-sofic group: the unit group of the binary Leavitt algebra. We exhibit a different source of examples (relying on the same technical criterion) which includes torsion-free groups.

math.GR↗

arXiv2026

Humanity's Last Exam

Benchmarks are important tools for tracking the rapid advancements in large language model (LLM) capabilities. However, benchmarks are not keeping pace in difficulty: LLMs now achieve over 90\% accuracy on popular benchmarks like MMLU, limiting informed measurement of state-of-the-art LLM capabilities. In response, we introduce Humanity's Last Exam (HLE), a multi-modal benchmark at the frontier of human knowledge, designed to be the final closed-ended academic benchmark of its kind with broad subject coverage. HLE consists of 2,500 questions across dozens of subjects, including mathematics, humanities, and the natural sciences. HLE is developed globally by subject-matter experts and consists of multiple-choice and short-answer questions suitable for automated grading. Each question has a known solution that is unambiguous and easily verifiable, but cannot be quickly answered via internet retrieval. State-of-the-art LLMs demonstrate low accuracy and calibration on HLE, highlighting a significant gap between current LLM capabilities and the expert human frontier on closed-ended academic questions. To inform research and policymaking upon a clear understanding of model capabilities, we publicly release HLE at https://lastexam.ai.

cs.LG↗

arXiv2026

Finiteness properties and Higman's rope trick

In 1961 Higman proved that every finitely generated recursively presented group embeds into a finitely presented group. In 2018 Leary proved that every finitely generated group embeds into a group of type $FP_2$. One naturally wonders whether analogous results hold for the higher finiteness properties $F_n$ and $FP_n$ ($3 \leq n \leq \infty$), i.e., whether every finitely generated recursively presented group embeds into a group of type $F_n$ and whether every finitely generated group embeds into a group of type $FP_n$. We prove that a positive answer to the $FP_n$ question that moreover preserves recursive presentability would imply a positive answer to the $F_n$ question. We also investigate the output groups from Higman's and Leary's proofs, which in both cases arise from the so-called ``Higman rope trick'', and find that they are never of type $FP_3(\mathbb{Q})$ (hence also never $FP_3$ nor $F_3$). Thus, any approach to the questions of higher finiteness properties must go via a different route than the Higman rope trick.

math.GR↗

arXiv2026

Ultrametric analogues of Ulam stability of groups

We study stability of metric approximations of countable groups with respect to groups endowed with ultrametrics, the main case study being a $p$-adic analogue of Ulam stability, where we take $GL_n(\mathbb{Z}_p)$ as approximating groups instead of $U(n)$. For finitely presented groups, the ultrametric nature implies equivalence of the pointwise and uniform stability problems, and the profinite one implies that the corresponding approximation property is equivalent to residual finiteness. Moreover, a group is uniformly stable if and only if its largest residually finite quotient is. We provide several examples of uniformly stable groups, including finite groups, virtually free groups, some groups acting on rooted trees, and certain lamplighter and (Generalised) Baumslag--Solitar groups. We construct a finitely generated group that is not uniformly stable. Finally, we prove and apply a (bounded) cohomological criterion for stability of a finitely presented group.

math.GR↗

arXiv2026

Bounded cohomology, quotient extensions, and hierarchical hyperbolicity

We call a central extension bounded if its Euler class is represented by a bounded cocycle. We prove that a bounded central extension of a hierarchically hyperbolic group (HHG) is still a HHG; conversely if a central extension is a HHG, then the extension is bounded, and under a further mild assumption the quotient is commensurable to a HHG. Motivated by questions on hierarchical hyperbolicity of quotients of mapping class groups, we therefore consider the general problem of determining when a quotient of a bounded central extension is still bounded, which we prove to be equivalent to an extendability problem for quasihomomorphisms. Finally, we show that quotients of the 4-strands braid group by suitable powers of a pseudo-Anosov are HHG, and in fact bounded central extensions of some HHG. We also speculate on how to extend the previous result to all mapping class groups.

math.GR↗

arXiv2026

Stability, approximable quotients, and higher property (T)

We construct a wealth of groups that are finitely presented, Frobenius stable, have property (T), but are very far from having property (T$_2$). Our method also shows that property (T$_2$) does not pass to quotients. A post-scriptum relates and comments on an experience this paper had with an AI benchmark.

math.GR↗

arXiv2026

An improved characterisation of inner automorphisms of groups

We show that every group $G$ embeds malnormally into a simple, complete co-Hopfian group $H$. This implies that a non-trivial endomorphism of $G$ extends to $H$ if and only if it is an inner automorphism, strengthening a theorem of Schupp and answering a question of Bergman.

math.GR↗

arXiv2026

Kazhdan groups of dimension $16$ with prescribed second $\ell^2$-Betti number

We construct a family of simple, lacunary hyperbolic groups with property $(T)$ that have rational cohomological dimension~$16$ and whose second $\ell^2$-Betti number can be prescribed to be any positive real. Moreover, we construct hyperbolic groups with property $(T)$ whose second $\ell^2$-Betti number can be prescribed to be any non-negative rational. Along the way, we present new constructions of measurably diverse finitely generated groups, and we prove that the second $\ell^2$-Betti number is far from being semi-continuous in the space of marked groups, even assuming good finiteness properties.

math.GR↗

arXiv2026

The Local Lifting Property, Property FD, and stability of approximate representations

We establish Kirchberg's Local Lifting Property and Lubotzky--Shalom's Property FD for classes of finitely generated groups of central importance in geometric and combinatorial group theory: $3$-manifold groups, limit groups, and certain one-relator groups and right-angled Artin groups. We deduce that such groups are very flexibly stable, with respect to normalized unitarily invariant norms. In the appendix, we show that these groups also have Kechris's property (E)MD, and hence are stable in finite actions, in the selse of Gohla--Thom. The exposition is made accessible to operator algebraists and group theorists alike.

math.GR↗

arXiv2026

COMPOSITE-Stem

AI agents hold growing promise for accelerating scientific discovery; yet, a lack of frontier evaluations hinders adoption into real workflows. Expert-written benchmarks have proven effective at measuring AI reasoning, but most at this stage have become saturated and only measure performance on constrained outputs. To help address this gap, we introduce COMPOSITE-STEM, a benchmark of 70 expert-written tasks in physics, biology, chemistry, and mathematics, curated by doctoral-level researchers. Our benchmark combines exact-match grading and criterion-based rubrics with an LLM-as-a-jury grading protocol, allowing more flexible assessment of scientifically meaningful outputs. Using an adapted multimodal Terminus-2 agent harness within the Harbor agentic evaluation framework, we evaluate four frontier models. The top-performing model achieves 21%, demonstrating that COMPOSITE-STEM captures capabilities beyond current agent reach. All tasks are open-sourced with contributor permission to support reproducibility and to promote additional research towards AI's acceleration of scientific progress in these domains.

cs.AI↗

arXiv2026

Hyperbolic actions of Thompson's group $F$ and generalizations

We study the poset of hyperbolic structures on Thompson's group $F$ and its generalizations $F_n$ for $n \geq 2$. The global structure of this poset is as simple as one would expect, with the maximal non-elementary elements being two quasi-parabolic actions corresponding to well-known ascending HNN-extension expressions of $F_n$. However, the local structure turns out to be incredibly rich, in stark contrast with the situation for the $T$ and $V$ counterparts. We show that the subposet of quasi-parabolic hyperbolic structures consists of two isomorphic posets, each of which contains uncountably many subposets of \emph{lamplike} structures, which can be described combinatorially in terms of certain hyperbolic structures on related lamplighter groups. Moreover, each of these subposets, as well as intersections and complements thereof, is very large, in that it contains a copy of the power set of the natural numbers. We also prove that these uncountably many uncountable subposets are not the entire picture, indeed there exists a copy of the power set of the natural numbers consisting entirely of non-lamplike structures. We also prove that this entire vast array of hyperbolic structures on $F_n$ collapses as soon as one takes a natural semidirect product with $\mathbb{Z}/2\mathbb{Z}$. These results are all proved via a detailed analysis of confining subsets, and along the way we establish a number of fundamental results in the theory of confining subsets of groups.

math.GR↗

arXiv2026

Abstract twisted Brin--Thompson groups

Given a group $G$ acting faithfully on a set $S$, one gets a simple group denoted $SV_G$, called a twisted Brin--Thompson group. In this paper we drop the faithfulness assumption, and get an abstract version of a twisted Brin--Thompson group $SV_G$. While the resulting group is not simple, since $SV_G$ surjects onto $SV_{G/\ker(G \curvearrowright S)}$, we prove that every proper normal subgroup of $SV_G$ lies in the kernel of this surjection, so $SV_G$ is ``relatively simple''. The advantage is that now we can prove that every finitely presented simple group embeds in a finitely presented abstract twisted Brin--Thompson group intersecting this kernel trivially. In particular, if the Boone--Higman conjecture is true, then so is a related conjectural characterization of groups with solvable word problem, arising purely in the world of twisted Brin--Thompson groups. We also prove a variety of additional results about abstract twisted Brin--Thompson groups, some of which are new even in the faithful case: they are all uniformly perfect, have property NL and property FW$_\infty$, are boundedly acyclic and $\ell^2$-invisible, and are $C^*$-simple as soon as they have trivial amenable radical. Along the way we formulate a new general criterion for $\ell^2$-invisibility that is interesting in its own right.

math.GR↗

arXiv2026

Profinite isomorphisms, stable commutator length, and fixed point properties

We construct Grothendieck pairs witnessing that the following are not profinite invariants: stable commutator length, quasimorphisms (answering a question of Echtler and Kammeyer), property NL (which obstructs actions on hyperbolic spaces), and property FW$_\infty$ (which obstructs actions on finite-dimensional CAT(0) cube complexes). We also recover that property FA and non-abelian free subgroups are not profinite invariants. The method combines Rips constructions with iterated group-theoretic Dehn filling on hyperbolic virtually special groups.

math.GR↗

arXiv2026

Quasihomomorphisms to real algebraic groups

A quasihomomorphism is a map that satisfies the homomorphism relation up to bounded error. Fujiwara and Kapovich proved a rigidity result for quasihomomorphisms taking values in discrete groups, showing that all quasihomomorphisms can be built from homomorphisms and sections of bounded central extensions. We study quasihomomorphisms with values in real linear algebraic groups, and prove an analogous rigidity theorem.

math.GR↗