Showing papers on "Free product published in 2023"

Product-free sets in the free group

Abstract: We prove that product-free sets of the free group over a finite alphabet have maximum density $1/2$ with respect to the natural measure that assigns total weight one to each set of irreducible words of a given length. This confirms a conjecture of Leader, Letzter, Narayanan and Walters. In more general terms, we actually prove that strongly $k$-product-free sets have maximum density $1/k$ in terms of the said measure. The bounds are tight.

A new class of a-T-menable free-by-free groups

Abstract: We prove that a semi-direct product of two finite rank free groups $F_k$ and $F_n$ such that $F_k$ acts on $F_n$ by polynomially growing automorphisms acts properly isometrically on a finite dimensional CAT(0) cube complex provided some finite-index unipotent subgroup of $F_k$ satisfies two supplementary properties. In particular any such group is a-T-menable in the sense of Gromov (equivalently satisfies the Haagerup property).

The structure and density of $k$-product-free sets in the free semigroup

Abstract: The free semigroup $\mathcal{F}$ over a finite alphabet $\mathcal{A}$ is the set of all finite words with letters from $\mathcal{A}$ equipped with the operation of concatenation. A subset $S$ of $\mathcal{F}$ is $k$-product-free if no element of $S$ can be obtained by concatenating $k$ words from $S$, and strongly $k$-product-free if no element of $S$ is a (non-trivial) concatenation of at most $k$ words from $S$. We prove that a $k$-product-free subset of $\mathcal{F}$ has upper Banach density at most $1/\rho(k)$, where $\rho(k) = \min\{\ell \colon \ell mid k - 1\}$. This confirms a conjecture of Ortega, Ru\'{e}, and Serra. We also determine the structure of the extremal $k$-product-free subsets for all $k otin \{3, 5, 7, 13\}$; a special case of this proves a conjecture of Leader, Letzter, Narayanan, and Walters. We further determine the structure of all strongly $k$-product-free sets with maximum density.

On equations in free monoids and semigroups with restrictions on solutions

Abstract: We study algorithmic problems for equations in free monoids and semigroups (equations in words and lengths) with additional restrictions on the solutions. It is proved that it is impossible to construct an algorithm that solves an arbitrary system of equations in words and lengths in a free monoid (free semigroup) of rank 2 with an additional constraint on the solution in the form that one of its components belongs to the language of balanced words or the equality of the projections of two components of the solution into a distinguished free generator to determine whether it has a solution. A similar result is obtained for systems of inequalities in words.

Classification of thin regular map representations of hypermaps

Abstract: There are two well known maps representations of hypermaps, namely the Walsh and the Vince map representations, being dual of each other. They correspond to normal subgroups of index two of a free product Γ = (C2×C2) * C2 which decompose as “elementary” free product C2 * C2 * C2. However Γ has three normal subgroups that decompose as “elementary" free product C2 * C2 * C2, the third of these sbgroups giving the less known petrie-path map representation. By relaxing the “elementary" free product condition to free product of rank 3, and under the extra condition “words of smaller length" on the generators, we prove that the number of map representations of hypermaps increases to 15 (up to a restrictedly dual), all of which described in this paper.

Interval Exchange Transformations groups. Free actions and dynamics of virtually abelian groups

Abstract: H\"older's theorem states that any group acting freely by circle homeomorphisms is abelian, this is no longer true for interval exchange transformations: we first give examples of free actions of non abelian groups. Then after noting that finitely generated groups acting freely by IET are virtually abelian, we classify the free actions of groups containing a copy of $\mathbb Z^2$, showing that they are ``conjugate" to actions in some specific subgroups $G_n$, namely $G_n \simeq ({\mathcal G}_2)^n \rtimes\mathcal S_n $ where ${\mathcal G}_2$ is the group of circular rotations seen as exchanges of $2$ intervals and $\mathcal S_n$ is the group of permutations of $\{1,...,n\}$ acting by permuting the copies of ${\mathcal G}_2$. We also study non free actions of virtually abelian groups and we obtain the same conclusion for any such group that contains a conjugate to a product of restricted rotations with disjoint supports and without periodic points. As a consequence, we provide examples of non virtually nilpotent subgroups of IETs. In particular, we show that the group generated by $f\in G_n$ periodic point free and $g otin G_n$ is not virtually nilpotent. Moreover, we exhibit examples of finitely generated non virtually nilpotent subgroups of IETs, some of them are metabelian and others are not virtually solvable.

Stability of amalgamated free products and HNN extensions

Abstract: We study stability of amalgamated free products and HNN extensions of stable groups over finite groups. We focus on operator norm stability, Hilbert-Schmidt stability and stability in permutations. We provide many new examples of stable (or flexibly stable) non-amenable groups.

Finite dimensional approximations of certain amalgamated free products of groups

Abstract: A group is called matricial field (MF) if it admits finite dimensional approximate unitary representations which are approximately faithful and approximately contained in the left regular representation. This paper provides a new class of MF groups by showing that given two amenable groups with a common normal subgroup, the amalgamated free product is MF.

Bounded projections to the $\mathcal{Z}$-factor graph

Abstract: Suppose $G$ is a free product $G = A_1 * A_2* \cdots * A_k * F_N$, where each of the groups $A_i$ is torsion-free and $F_N$ is a free group of rank $N$. Let $\mathcal{O}$ be the deformation space associated to this free product decomposition. We show that the diameter of the projection of the subset of $\mathcal{O}$ where a given element has bounded length to the $\mathcal{Z}$-factor graph is bounded, where the diameter bound depends only on the length bound. This relies on an analysis of the boundary of $G$ as a hyperbolic group relative to the collection of subgroups $A_i$ together with a given non-peripheral cyclic subgroup. The main theorem is new even in the case that $G = F_N$, in which case $\mathcal{O}$ is the Culler-Vogtmann outer space. In a future paper, we will apply this theorem to study the geometry of free group extensions.

A characterization of quasi-transitive tree-like inverse graphs

Abstract: We extend the characterization of context-free groups of Muller and Schupp in two ways. We first show that for a quasi-transitive inverse graph $\Gamma$, being quasi-isometric to a tree, is equivalent to being context-free, which in turn is equivalent to having the automorphism group $Aut(\Gamma)$ that is virtually free. As a consequence of this characterization, we solve a weaker version of a conjecture of T. Brough which also extends Muller and Schupp's result to the class of groups that are virtually finitely generated subgroups of direct product of free groups.

One-endedness of outer automorphism groups of free products of finite and cyclic groups

Abstract: In a previous paper, we showed that the group of outer automorphisms of the free product of two nontrivial finite groups with an infinite cyclic group has infinitely many ends, despite being of virtual cohomological dimension two. The main result of this paper is that aside from this exception, having virtual cohomological dimension at least two implies the outer automorphism group of a free product of finite and cyclic groups is one ended. As a corollary, the outer automorphism group of the free product of four finite groups or the free product of a single finite group with a free group of rank two is a virtual duality group of dimension two, in contrast with the above example. We also prove that groups in this family are semistable at infinity (or at each end). Our proof is inspired by methods of Vogtmann, applied to a complex first studied in another guise by Krsti\'c and Vogtmann.

Product-free sets in the free group

Abstract: We prove that product-free sets of the free group over a finite alphabet have maximum density $1/2$ with respect to the natural measure that assigns total weight one to each set of irreducible words of a given size. This confirms a conjecture of Leader, Letzter, Narayanan and Walters. In more general terms, we actually prove that strongly $k$-product-free sets have maximum density $1/k$ in terms of the said measure.

Note on the conjugacy classes of elements and their centralizers for the free product of two groups

Abstract: We describe the conjugacy classes of the elements of the free product of two groups and their centralizers and, as a consequence, we correct the calculation of the cyclic and periodic cyclic homology of the group ring of the free product of two groups given in a previous paper.

A Study of Free Groups

Abstract: The purpose of this paper is to develop and discuss a particular type of group, the free group. First the construction of the group is given, followed by the proof that it is in fact a group and a discussion of its properties. Included in this discussion are several formal proofs of some of its properties, and examples of free groups. One of these properties relates free groups to the theory of groups as a whole J and it states: \Every group is isomorphic to a factor group of a free group\ [3, p.128]. Closely related to this property is the topic of defining relations, which is then developed, discussed and exemplified.

Free products from spinning and rotating families

Abstract: The far-reaching work of Dahmani–Guirardel–Osin (2017) and recent work of Clay– Mangahas–Margalit (2021) provide geometric approaches to the study of the normal closure of a subgroup (or a collection of subgroups) in an ambient group $G$. Their work gives conditions under which the normal closure in $G$ is a free product. In this paper we unify their results and simplify and significantly shorten the proof of the theorem of Dahmani–Guirardel–Osin (2017).