Related papers: On what finitely generated (left-orderable) simple…

The isomorphism problem for infinite finitely presented groups is probably the hardest among standard algorithmic problems in group theory. Classes of groups where it has been completely solved are nilpotent groups, hyperbolic groups, and…

We analyze the classification problem for finitely generated orderable groups from the viewpoint of descriptive set theory. We analyze the standard Borel space of finitely generated left-orderable groups, and the subspace of finitely…

In this note we show that groups with definable generics in a separably closed valued of finite imperfection degree can be embedded into groups definable in their algebraic closure.

We augment the body of existing results on embedding finite semigroups of a certain type into 2-generator finite semigroups of the same type. The approach adopted applies to finite semigroups the idempotents of which form a band and also to…

This is an almost self-contained monograph (containing some new results) on left-orderable groups which mostly rely on dynamical and probabilistic aspects, but also on geometric, combinatorial, analytic, and topological ones. This new…

This is a draft of a book submitted for publication by the AMS. Its theme is the remarkable interplay, accelerating in the last few decades, between topology and the theory of orderable groups, with applications in both directions. It…

In $2019$ Hyde and the second author constructed the first family of finitely generated, simple, left orderable groups. We prove that these groups are not finitely presentable, non-inner amenable, don't have Kazhdan's property $(T)$ (yet…

We study the relative growth of finitely generated subgroups in finitely generated groups, and the corresponding distortion function of the embeddings. We explore which functions are equivalent to the relative growth functions and…

This paper gives a quick overview of the author's recent result that all finitely presented groups are QSF.

The purpose of this paper is to explore the concept of localization, which comes from homotopy theory, in the context of finite simple groups. We give an easy criterion for a finite simple group to be a localization of some simple subgroup…

This article presents the construction of finitely generated branch groups with uncountably many maximal subgroups using embedding techniques. This addresses a question posed by Grigorchuk.

The present survey aims at being a list of Conjectures and Problems in an area of model-theoretic algebra wide open for research, not a list of known results. To keep the text compact, it focuses on structures of finite Morley rank,…

We demonstrate the existence of a family of finitely generated subgroups of Richard Thompson's group $F$ which is strictly well-ordered by the embeddability relation in type $\epsilon_0 +1$. All except the maximum element of this family…

We propose an algorithm which for any recursive group $G$, given by its effectively enumerable generators and recursively enumerable relations, outputs an explicit embedding of $G$ into a finitely presented group directly written by its…

In the context of Higman embeddings of recursive groups into finitely presented groups we suggest an algorithm which uses Higman operations to explicitly constructs the specific recursively enumerable sets of integer sequences arising…

We show that there is an order-preserving embedding of the additive group of rational numbers $\mathbb{Q}$ into a 2-generator group $G$. The group $G$ can be chosen to be a solvable group $G$ of length 3, which is a minimal result in the…

It is shown that a closed solvable subgroup of a connected Lie group is compactly generated. In particular, every discrete solvable subgroup of a connected Lie group is finitely generated. Generalizations to locally compact groups are…

Let $F$ be a finitely generated free group. We present an algorithm such that, given a subgroup $H\leqslant F$, decides whether $H$ is the fixed subgroup of some family of automorphisms, or family of endomorphisms of $F$ and, in the…

We report on recent progress concerning the relationship that exists between the algebraic structure of a finite group and certain features of its class-size prime graph.

A set A of natural numbers is finitely embeddable in another such set B if every finite subset of A has a rightward translate that is a subset of B. This notion of finite embeddability arose in combinatorial number theory, but in this paper…