Related papers: The Absolute of finitely generated groups: I. Comm…

We initiate the study of a measurable analogue of small topological full groups that we call $\mathrm L^1$ full groups. These groups are endowed with a Polish group topology which admits a natural complete right invariant metric. We mostly…

In 2022, using methods from ergodic theory, Kra, Moreira, Richter, and Robertson resolved a longstanding conjecture of Erd\H{o}s about sumsets in large subsets of the natural numbers. In this paper, we extend this result to several…

A B-group is a group such that all its minimal generating sets (with respect to inclusion) have the same size. We prove that the class of finite B-groups is closed under taking quotients and that every finite B-group is solvable. Via a…

The main goal of this note is to suggest an algebraic approach to the quasi-isometric classification of partially commutative groups (alias right-angled Artin groups). More precisely, we conjecture that if the partially commutative groups…

The first part of the paper centers in the study of embeddability between partially commutative groups. In [KK], for a finite simplicial graph $\Gamma$, the authors introduce an infinite, locally infinite graph $\Gamma^e$, called the…

We lay the foundations for the study of relatively quasiconvex subgroups of relatively hyperbolic groups. These foundations require that we first work out a coherent theory of countable relatively hyperbolic groups (not necessarily finitely…

The commuting graph of a group $G$ is the simple undirected graph whose vertices are the non-central elements of $G$ and two distinct vertices are adjacent if and only if they commute. It is conjectured by Jafarzadeh and Iranmanesh that…

For a topological group $G$ let $E_{\textsf{com}}(G)$ be the total space of the universal transitionally commutative principal $G$-bundle as defined by Adem--Cohen--Torres-Giese. So far this space has been most studied in the case of…

In this article we prove some previously announced results about metric ultraproducts of finite simple groups. We show that any non-discrete metric ultraproduct of alternating or special linear groups is a geodesic metric space. For more…

We investigate the computational complexity for determining various properties of a finite transformation semigroup given by generators. We introduce a simple framework to describe transformation semigroup properties that are decidable in…

This paper develops a theory for completely random measures in the framework of free probability. A general existence result for free completely random measures is established, and in analogy to the classical work of Kingman it is proved…

We consider the class $\mathcal{R}$ of finitely generated toral relatively hyperbolic groups. We show that groups from $\mathcal{R}$ are commutative transitive and generalize a theorem proved by Benjamin Baumslag to this class. We also…

This paper has two main parts. In the first part we develop an elementary coordinatization for any nilpotent group $G$ taking exponents in a binomial principal ideal domain (PID) $A$. In case that the additive group $A^+$ of $A$ is finitely…

In this paper, we present a new technique for studying the dynamics of a finitely generated rational semigroup. Such a semigroup can be associated naturally to a certain holomorphic correspondence on $\mathbb{P}^1$. Then, results on the…

Commutative semirings with divisible additive semigroup are studied. We show that an additively divisible commutative semiring is idempotent, provided that it is finitely generated and torsion. In case that a one-generated additively…

We obtain upper bounds on the composition length of a finite permutation group in terms of the degree and the number of orbits, and analogous bounds for primitive, quasiprimitive and semiprimitive groups. Similarly, we obtain upper bounds…

Given a finite graph G there is a corresponding group given by the presentation with generators the vertices of G and a relation [x,y]=1 for generators x and y precisely when (x,y) is an edge of G. Such groups are known as partially…

The dead-end depth of an element g of a group G, with respect to a generating set A is the distance from g to the complement of the radius $d_A(1,g)$ closed ball, in the word metric $d_A$ defined with respect to A. We exhibit a finitely…

A group element is called generalized torsion if a finite product of its conjugates is equal to the identity. We show that in a finitely generated abelian-by-finite group, an element is generalized torsion if and only if its image in the…

A group element is called a generalized torsion if a finite product of its conjugates is equal to the identity. We prove that in a nilpotent or FC-group, the generalized torsion elements are all torsion elements. Moreover, we compute the…