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André Carvalho

Publications and source records attributed to André Carvalho.

At least 19 recordsLinked to original sources
arXiv2026

A non-plain virtually free group with a strictly 2-geodetic Cayley graph

We construct an explicit infinite virtually free group $G=C_4\ast_{C_2}D_3$ with a finite inverse-closed generating set whose Cayley graph is $2$-geodetic but not geodetic. More precisely, we describe the full language of geodesics and show that every element of $G$ has a unique geodesic representative except for a single element, which has exactly two. This gives a counterexample to the conjecture that an infinite group admitting a $k$-geodetic Cayley graph must admit a geodetic Cayley graph with respect to the same generating set. We further prove that $G$ is freely indecomposable and hence non-plain. Consequently, Shapiro's conjecture that every geodetic group is plain and the conjecture that every hyperbolic $k$-geodetic group is geodetic cannot both hold.

math.GR↗
arXiv2026

Green languages and growth of free inverse monoids

Motivated by recent work on conjugacy languages in groups, we introduce a general framework for studying \emph{languages of representatives} associated to equivalence relations on finitely generated semigroups. After developing this notion in full generality, we particularize it to Green's relations and obtain the corresponding Green languages, which form the central objects of this article. For free inverse monoids, we describe these languages via Munn trees and establish several structural relations among them, including characterizations of the $\mathcal{D}$-, $\mathcal{R}$-, $\mathcal{L}$-, and $\mathcal{H}$-languages. We then investigate the formal language-theoretic complexity of these Green languages. For free inverse monoids of rank at least two, we show that $\mathcal{H}$-Geo is context-free and co-context-free but not regular, while $\mathcal{R}$-Geo, $\mathcal{L}$-Geo, and $\mathcal{D}$-Geo are neither context-free nor co-context-free. In contrast, in the monogenic case, $\mathcal{D}$-Geo is regular and the remaining Green languages are deterministic context-free. We further relate these results to the ShortLex language and the growth of free inverse monoids, proving in particular that $(\text{FIM}_X,X)$ is growth tight. The article concludes with a collection of open problems and directions for future research.

math.GR↗
arXiv2026

Standard morphisms and Pythagorean triples

Let $m\geq 1$, let $f:\mathbb N\to\mathbb Z/m\mathbb Z$ be a standard morphism and let $T(m)$ be the least integer $N$ such that every such $f$ admits a primitive monochromatic Pythagorean triple with hypotenuse at most $N$. The aim of this note is to prove that every standard morphism has infinitely many identity-valued Pythagorean triples and infinitely many primitive monochromatic Pythagorean triples. Thus the qualitative part of Problem~4.3 of Eliahou, Fromentin, Marion-Poty and Robilliard is solved for every $m$. Moreover, $T(m)$ is finite, the morphism $n\mapsto v_3(n)\pmod m$ has least possible hypotenuse $(9^m+1)/2$ and $T(d)\leq T(m)$ when $d\mid m$.

math.CO↗
arXiv2026

Conjugacy languages in free inverse monoids

We initiate the study of conjugacy languages in free inverse monoids. Motivated by the notion of conjugacy languages in groups and the study of conjugacy in semigroups, we introduce the language of shortest representatives of conjugacy classes and study it for free inverse monoids of rank at least $2$ under the natural notion of conjugacy. We show that, contrary to the free group case, where this language is regular, in a free inverse monoid it is neither context-free nor co-context-free. In the monogenic case, this language is context-free. We show that this non-context-freeness comes from elements with nontrivial conjugacy classes. We define an equivalence relation as follows: all elements with a nontrivial conjugacy class are related and elements with trivial conjugacy class are only related to themselves. We call this relation $\mathrm{UConj}$. We show that the language consisting of geodesics representing elements whose conjugacy class is trivial is context-free by providing an explicit context-free grammar generating it. For groups, we show that the language of minimal representatives of $\mathrm{UConj}$ classes is regular if the group is hyperbolic and for right-angled Artin groups with the standard generating set, it is piecewise testable. For virtually abelian groups, we show that there is a generating set for which this language is piecewise excluding and exhibit an example of a virtually abelian group admitting a generating set for which this language is not regular.

math.GR↗
arXiv2026

The Post Correspondence Problem for free groups is undecidable

We prove that the Post Correspondence Problem for finitely generated free groups is undecidable, even when one of the two homomorphisms is injective and has finite-index image. This resolves a longstanding open problem in algorithmic group theory. The proof proceeds through a connection with finite-state transducers. Given a cyclic tag system $\mathcal C$, we effectively construct a finite partial deterministic inverse transducer $\mathcal T_{\mathcal C}$ whose fixed-point set is nontrivial if and only if $\mathcal C$ halts. We then associate to any such transducer two homomorphisms $g,h\colon F_Y\longrightarrow F_A,$ with $h$ injective, such that their equalizer is nontrivial precisely when the transducer has a nontrivial fixed loop. As an immediate consequence, the rank of these equalizers cannot be computed in general, answering a question posed by Stallings in 1984. We further prove that there is no algorithm which decides whether the fixed subgroup of a virtual endomorphism of a finitely generated free group is trivial. Finally, we apply the main result to show that the stabilizer problem is undecidable for free subgroups of $\operatorname{SL}_4(\mathbb Z)$, and that the upper-right-corner problem is undecidable for free subgroups of $\operatorname{SL}_5(\mathbb Z)$, even when the given generators are promised to form a free basis, improving on recent results of Breuillard and Kocharyan.

math.GR↗
arXiv2026

Rational dynamics of a prime-representing map

We study the rational dynamics of the map $\mathcal{T}(x)=\lfloor x\rfloor(1+\{x\})$, which appears in the recursive construction of the prime-representing constant of Fridman, Garbulsky, Glecer, Grime and Florentin. For a rational number $x\geq 2$ with denominator $M$, we define its order to be the least non-negative integer $n$ such that $\mathcal{T}^n(x)$ is an integer, if such an $n$ exists, and ask whether every rational number has finite order. For each \(n\), we prove that the reduced fractions \(a/M\) of exact order \(n\) are described by residue classes of \(a\) modulo \(M^{n+1}\), and give a recurrence for the number $A(n,M)$ of residue classes of exact order $n$. We then show that for each fixed denominator the fractions of finite order have natural density one among all reduced fractions with that denominator, which implies in particular that there is no infinite arithmetic progression of rational numbers of infinite order. We also give an explicit family of fractions of prescribed order for every denominator, and fully characterize the case $M=2$.

math.NT↗
arXiv2026

On fixed points and equalizers of injective endomorphisms of the free group at infinity

We study equalizers and fixed points of monomorphisms of free groups at infinity. We show that the action of the equalizer of two monomorphisms on the regular points of the equalizer at infinity has finitely many orbits, showing that the equalizer at infinity is, in some sense, finitely generated and generalizing a previous result of Cooper about fixed points. We additionally show that it is decidable whether an automorphism of a free group has a nontrivial fixed point at infinity. The same result is shown for monomorphisms satisfying the condition of being almost length-increasing. We remark that almost length-increasing monomorphisms are generic among endomorphisms of a free group. We also prove that being almost length-increasing is a decidable condition. We end the paper with several open problems arising from this work.

math.GR↗
arXiv2026

A language-theoretic approach to study the density of subsets in free groups

In this paper, we study the density of subsets of nonabelian free groups using relative densities of languages. We start by proving some basic properties about the density of a language $L_1$ relative to another language $L_2$ containing $L_1$. We then focus on the case where $L_2$ is the language of freely reduced words over an alphabet and prove an analogue of the Infinite Monkey Theorem for this language. This result, obtained as a corollary of a broader theorem on irreducible subshifts of finite type, allows for a language-theoretic characterization of rational subsets with positive density. As a consequence, we obtain a language-theoretic proof that the automorphic orbit of an element of a nonabelian free group has natural density zero, which generalizes a result by Burillo and Ventura concerning the density of primitive elements of free groups. We then describe rational subsets of free groups of positive density and explore in depth the case of finitely generated subgroups. We prove that if the rational subset is a subgroup, then it has positive density if and only if it has finite index and characterize those for which there is convergence. In cases where convergence fails due to parity constraints, we show that the density of the subset always exhibits weak convergence and that the average of its supremum and infimum densities converges to the expected value.

math.GR↗
arXiv2025

International AI Safety Report 2025: Second Key Update: Technical Safeguards and Risk Management

This second update to the 2025 International AI Safety Report assesses new developments in general-purpose AI risk management over the past year. It examines how researchers, public institutions, and AI developers are approaching risk management for general-purpose AI. In recent months, for example, three leading AI developers applied enhanced safeguards to their new models, as their internal pre-deployment testing could not rule out the possibility that these models could be misused to help create biological weapons. Beyond specific precautionary measures, there have been a range of other advances in techniques for making AI models and systems more reliable and resistant to misuse. These include new approaches in adversarial training, data curation, and monitoring systems. In parallel, institutional frameworks that operationalise and formalise these technical capabilities are starting to emerge: the number of companies publishing Frontier AI Safety Frameworks more than doubled in 2025, and governments and international organisations have established a small number of governance frameworks for general-purpose AI, focusing largely on transparency and risk assessment.

cs.CY↗
arXiv2025

Conjugacy languages and conjugacy growth relative to subsets of groups

In this paper, we explore conjugacy languages when the base problem is the generalized conjugacy problem (with constraints): given $g\in G$ and $U\subset G$, does $g$ have a conjugate in $U$ (with conjugators in a certain subset)? To do so, for subsets $U,V\subseteq G$, we define the corresponding languages $\text{ConjGeo(U,V)}$, $\text{CycGeo(U)}$, $\text{ConjSL(U)}$ and $\text{ConjMinLenSL(U,V)}$, following the previously studied cases where $U=V=G$. Our results cover several classes of groups: for free groups, we prove that $\text{ConjGeo(U,V)}$ and $\text{ConjMinLenSL(U,V)}$ are regular if $U$ and $V$ are rational subsets; for hyperbolic groups, we show that if $L$ is a regular language of geodesics and $U$ is the subsets represented by it, then $\text{ConjGeo(U)}$ and $\text{ConjMinLenSL(U)}$ are regular; for virtually cyclic groups, we show that $\text{ConjSL(U)}$ is regular if $U$ is rational; and, for virtually abelian groups, we prove that $\text{ConjGeo(U)}$ belongs to a certain class of languages $\C$ when the language of words representing elements of $U$ also belongs to $\C$. We also define relative conjugacy growth and show that its behavior can be heavily dependent on the choice of subset.

math.GR↗
arXiv2025

International AI Safety Report 2025: First Key Update: Capabilities and Risk Implications

Since the publication of the first International AI Safety Report, AI capabilities have continued to improve across key domains. New training techniques that teach AI systems to reason step-by-step and inference-time enhancements have primarily driven these advances, rather than simply training larger models. As a result, general-purpose AI systems can solve more complex problems in a range of domains, from scientific research to software development. Their performance on benchmarks that measure performance in coding, mathematics, and answering expert-level science questions has continued to improve, though reliability challenges persist, with systems excelling on some tasks while failing completely on others. These capability improvements also have implications for multiple risks, including risks from biological weapons and cyber attacks. Finally, they pose new challenges for monitoring and controllability. This update examines how AI capabilities have improved since the first Report, then focuses on key risk areas where substantial new evidence warrants updated assessments.

cs.CY↗
arXiv2025

On supratopologies, normalized families and Frankl conjecture

We introduce some generalized topological concepts to deal with union-closed families, and show that one can reduce the proof of Frankl's conjecture to some families of so-called supratopological spaces. We prove some results on the structure of normalized families, presenting a new way of reducing such a family to a smaller one using dual families. Applying our reduction method, we prove a refinement of a conjecture originally proposed by Poonen. Finally, we show that Frankl's Conjecture holds for the class of families obtained from successively applying the reduction process to a power set.

math.CO↗
arXiv2025

On the dynamics of endomorphisms of the direct product of two free groups

We prove that Brinkmann's problems are decidable for endomorphisms of $F_n\times F_m$: given $(x,y),(z,w)\in F_n\times F_m$ and $Φ\in \text{End}(F_n\times F_m)$, it is decidable whether there is some $k\in \mathbb{N}$ such that $(x,y)Φ^k=(z,w)$ (or $(x,y)Φ^k\sim(z,w)$). We also prove decidability of a two-sided version of Brinkmann's conjugacy problem for injective endomorphisms which, from the work of Logan, yields a solution to the conjugacy problem in ascending HNN-extensions of $F_n\times F_m$. Finally, we study the dynamics of automorphisms of $F_n\times F_m$ at the infinity, proving that that their dynamics at the infinity is asymptotically periodic, as occurs in the free and free-abelian times free cases.

math.GR↗
arXiv2025

On linguistic subsets of groups and monoids

We study subsets of groups and monoids defined by language-theoretic means, generalizing the classical approach to the word problem. We expand on results by Herbst from 1991 to a more general setting, and for a class of languages $\mathbf{C}$ we define the classes of $\mathbf{C}^\forall$-flat and $\mathbf{C}^\exists$-flat groups. We prove several closure results for these classes of groups, prove a connection with the word problem, and characterize $\mathbf{C}^\forall$-flat groups for several classes of languages. In general, we prove that the class of $\mathbf{C}^\forall$-flat groups is a strict subclass of the class of groups with word problem in $\mathbf{C}$, including for the class $\mathbf{REC}$ of recursive languages, for which $\mathbf{C}^\forall$-flatness for a group resp. monoid is proved to be equivalent to the decidability of the subgroup membership problem resp. the submonoid membership problem. We provide a number of examples, including the Tarski monsters of Ol'shanskii, showing the difficulty of characterizing $\mathbf{C}^\exists$-flat groups. As an application of our general methods, we also prove in passing that if $\mathbf{C}$ is a full semi-$\mathrm{AFL}$, then the class of epi-$\mathbf{C}$ groups is closed under taking finite index subgroups. This answers a question recently posed by Al Kohli, Bleak & Elliott.

math.GR↗
arXiv2025

Orbit problems for free-abelian times free groups and related families

We prove that the Brinkmann Problems (BrP & BrCP) and the twisted-conjugacy Problem (TCP) are decidable for any endomorphism of a free-abelian times free (FATF) group Fn x Z^m. Furthermore, we prove the decidability of the two-sided Brinkmann conjugacy problem (2BrCP) for monomorphisms of FATF groups (and combine it with TCP) to derive the decidability of the conjugacy problem for ascending HNN extensions of FATF groups.

math.GR↗
arXiv2025

On generalized (twisted) conjugacy separability of some extensions of groups

We introduce separability properties corresponding to generalized versions of the conjugacy, twisted conjugacy, Brinkmann and Brinkmann's conjugacy problems and how they relate when finite and cyclic extensions of groups are taken. In particular, we prove that some (concrete) generalizations of twisted conjugacy separability of a group $G$ with respect to virtually inner automorphisms are equivalent to some (concrete) generalizations of the conjugacy problem in finite extensions of $G$. Similarly, (generalized) conjugacy separability in cyclic extensions of $G$ implies (generalized) twisted conjugacy and Brinkmann's conjugacy separability in $G$. Applications include results in free, virtually abelian, virtually polycyclic groups and a proof that virtually free times free groups are conjugacy separable.

math.GR↗
arXiv2024

Geodesic languages for rational subsets and conjugates in virtually free groups

We prove that a subset of a virtually free group is rational if and only if the language of geodesic words representing its elements (in any generating set) is rational and that the language of geodesics representing conjugates of elements in a rational subset of a virtually free group is context-free. As a corollary, the doubly generalized conjugacy problem is decidable for rational subsets of finitely generated virtually free groups: there is an algorithm taking as input two rational subsets $K_1$ and $K_2$ of a virtually free group that decides whether there is one element of $K_1$ conjugate to an element of $K_2$. For free groups, we prove that the same problem is decidable with rational constraints on the set of conjugators.

math.GR↗