arXiv subjects
Publications and source records attributed to Francesco Matucci.
Generalizing works of D'Angeli and Donno, we describe, starting from an infinite sequence over $r$ letters with $r \neq 4i$ and $i \in \mathbb{N}$, a sequence of pointed finite graphs. We study the pointed Gromov-Hausdorff limit graphs giving a description of isomorphim classes in terms of dihedral groups and providing insights on the horofunction boundaries in terms of Busemann and non-Busemann points.
We are interested in the subgroup membership problem in groups acting on rooted $d$-regular trees and a natural class of subgroups, the stabilisers of infinite rays emanating from the root. These rays, which can also be viewed as infinite words in the alphabet with d letters, form the boundary of the tree. Stabilisers of infinite rays are not finitely generated in general, but if the ray is computable, the membership problem is well posed and solvable. The main result of the paper is that, for bounded automata groups, the membership problem in the stabiliser of any ray that is eventually periodic as an infinite word, forms an ET0L language that is constructable. The result is optimal in the sense that, in general, the membership problem for the stabiliser of an infinite ray in a bounded automata group cannot be context-free. As an application, we give a recursive formula for the associated generating function, aka the Green function, on the corresponding infinite Schreier graph.
Starting from context-free inverse graphs, we introduce a new class of groups and study their structural properties. We establish closure properties, show that their co-word problems are context-free, analyze torsion elements, and realize them as subgroups of the asynchronous rational group. Context-freeness is preserved under a generalized free product of graphs, and using this construction we provide examples of groups that are not residually finite or not poly-context-free, making them relevant for testing the Lehnert and Brough conjectures. Moreover, we investigate how small local modifications of a graph affect the global structure of the transition group, showing that for locally quasi-transitive graphs with infinite orbits, the transition group decomposes into a highly structured quotient by a bounded torsion subgroup, showing strong global constraints induced by local graph properties.
We prove finiteness properties for groups of homeomorphisms that have finitely many "singular points", and we describe the normal structure of such groups. As an application, we prove that every countable abelian group can be embedded into a finitely presented simple group, verifying the Boone-Higman conjecture for countable abelian groups. Indeed, we describe a specific 2-generated, $\mathrm{F}_\infty$ simple group $V\mathcal{A}$ of homeomorphisms of the Cantor set that contains every countable abelian group. As a second application, we prove that if $G$ is a bounded automata group then the associated R\"over-Nekrashevych groups $V_{d,r}G$ have type $\mathrm{F}_\infty$, verifying a conjecture of Nekrashevych for a large class of contracting self-similar groups. Among others, this result applies to R\"{o}ver-Nekrashevych groups associated to Gupta-Sidki groups and the basilica group.
We give a short proof that every contracting self-similar group embeds into a finitely presented simple group. In particular, any contracting self-similar group embeds into the corresponding R\"over--Nekrashevych group, and this in turn embeds into one of the twisted Brin--Thompson groups introduced by the first author and Matthew Zaremsky. The proof here is a simplification of a more general argument given by the authors, Collin Bleak, and Matthew Zaremsky for contracting rational similarity groups.
The 1973 Boone-Higman conjecture predicts that every finitely generated group with solvable word problem embeds in a finitely presented simple group. In this paper, we show that hyperbolic groups satisfy this conjecture, that is, each hyperbolic group embeds in some finitely presented simple group. This shows that the conjecture holds in the "generic" case for finitely presented groups. Our key tool is a new family of groups, which we call "rational similarity groups (RSGs)", that is interesting in its own right. We prove that every hyperbolic group embeds in a full, contracting RSG, and every full, contracting RSG embeds in a finitely presented simple group, thus establishing the result. Another consequence of our work is that all contracting self-similar groups satisfy the Boone-Higman conjecture.
A conjecture of Boone and Higman from the 1970's asserts that a finitely generated group $G$ has solvable word problem if and only if $G$ can be embedded into a finitely presented simple group. We comment on the history of this conjecture and survey recent results that establish the conjecture for many large classes of interesting groups.
A group $G$ is integrable if it is isomorphic to the derived subgroup of a group $H$; that is, if $H'\simeq G$, and in this case $H$ is an integral of $G$. If $G$ is a subgroup of $U$, we say that $G$ is integrable within $U$ if $G=H'$ for some $H\leq U$. In this work we focus on two problems posed in [1]. We classify the almost-simple finite groups $G$ that are integrable, which we show to be equivalent to those integrable within $\mathrm{Aut}(S)$, where $S$ is the socle of $G$. We then classify all $2$-homogeneous subgroups of the finite symmetric group $S_n$ that are integrable within $S_n$.
In a remarkable theorem, M. Rubin proved that if a group $G$ acts in a locally dense way on a locally compact Hausdorff space $X$ without isolated points, then the space $X$ and the action of $G$ on $X$ are unique up to $G$-equivariant homeomorphism. Here we give a short, self-contained proof of Rubin's theorem, using equivalence classes of ultrafilters on a poset to reconstruct the points of the space $X$.
We investigate stabilizers of finite sets of rational points in Cantor space for the Higman-Thompson groups $V_{n,r}$. We prove that the pointwise stabilizer is an iterated ascending HNN extension of $V_{n,q}$ for any $q\geq 1$. We also prove that the commutator subgroup of the pointwise stabilizer is simple, and we compute the abelianization. Finally, for each $n$ we classify such pointwise stabilizers up to isomorphism.
We prove Thompson's group $F$ has quadratic conjugator length function. That is, for any two conjugate elements of $F$ of length $n$ or less, there exists an element of $F$ of length $O(n^2)$ that conjugates one to the other. Moreover, there exist conjugate pairs of elements of $F$ of length at most $n$ such that the shortest conjugator between them has length $\Omega(n^2)$. This latter statement holds for $T$ and $V$ as well.
An $integral$ of a group $G$ is a group $H$ whose commutator subgroup is isomorphic to $G$. This paper continues the investigation on integrals of groups started in the work arXiv:1803.10179. We study: (1) A sufficient condition for a bound on the order of an integral for a finite integrable group and a necessary condition for a group to be integrable. (2) The existence of integrals that are $p$-groups for abelian $p$-groups, and of nilpotent integrals for all abelian groups. (3) Integrals of (finite or infinite) abelian groups, including nilpotent integrals, groups with finite index in some integral, periodic groups, torsion-free groups and finitely generated groups. (4) The variety of integrals of groups from a given variety, varieties of integrable groups and classes of groups whose integrals (when they exist) still belong to such a class. (5) Integrals of profinite groups and a characterization for integrability for finitely generated profinite centreless groups. (6) Integrals of Cartesian products, which are then used to construct examples of integrable profinite groups without a profinite integral. We end the paper with a number of open problems.
We observe that the group of all lifts of elements of Thompson's group $T$ to the real line is finitely presented and contains the additive group $\mathbb{Q}$ of the rational numbers. This gives an explicit realization of the Higman embedding theorem for $\mathbb{Q}$, answering a Kourovka notebook question of Martin Bridson and Pierre de la Harpe.
Monod introduced in [14] a family of Thompson-like groups which provides natural counterexamples to the von Neumann-Day conjecture. We construct a characterization of conjugacy and invariant and use them to compute centralizers in one group of this family.
In this work we study automorphisms of synchronous self-similar groups, the existence of extensions to automorphisms of the full group of automorphisms of the infinite rooted tree on which these groups act on. When they do exist, we obtain conditions for the continuity of such extensions with respect to the depth metric, but we also construct examples of groups where such extensions do not exist. We study the case of the lamplighter group $\mathcal{L}_k = \mathbb{Z}_k \wr \mathbb{Z}$
An $integral$ of a group $G$ is a group $H$ whose derived group (commutator subgroup) is isomorphic to $G$. This paper discusses integrals of groups, and in particular questions about which groups have integrals and how big or small those integrals can be. Our main results are: (1) If a finite group has an integral, then it has a finite integral. (2) A precise characterization of the set of natural numbers $n$ for which every group of order $n$ is integrable: these are the cubefree numbers $n$ which do not have prime divisors $p$ and $q$ with $q\mid p-1$. (3) An abelian group of order $n$ has an integral of order at most $n^{1+o(1)}$, but may fail to have an integral of order bounded by $cn$ for constant $c$. (4) A finite group can be integrated $n$ times (in the class of finite groups) if and only if it is the central product of an abelian group and a perfect group. There are many other results on such topics as centreless groups, groups with composition length $2$, and infinite groups. We also include a number of open problems.
We prove that all Gromov hyperbolic groups embed into the asynchronous rational group defined by Grigorchuk, Nekrashevych and Sushchanski\u{i}. The proof involves assigning a system of binary addresses to points in the Gromov boundary of $G$, and proving that elements of $G$ act on these addresses by transducers. These addresses derive from a certain self-similar tree of subsets of $G$, whose boundary is naturally homeomorphic to the horofunction boundary of $G$.
We prove that the asynchronous rational group defined by Grigorchuk, Nekrashevych, and Sushchanskii is simple and not finitely generated. Our proofs also apply to certain subgroups of the rational group, such as the group of all rational bilipschitz homeomorphisms.