Related papers: On the isolated points in the space of groups
The study of localizations of groups has concentrated on group theoretic properties which are preserved by localization. In this paper we look at finitely generated soluble groups and determine when the local groups associated with them are…
Let $\mathcal G$ denote the space of finitely generated marked groups. For any finitely generated group $G$, we construct a continuous, injective map $f$ from the space of subgroups $Sub(G)$ to $\mathcal G$ that sends conjugate subgroups to…
We give a characterization of limits of dihedral groups in the space of finitely generated marked groups. We also describe the topological closure of dihedral groups in the space of marked groups on a fixed number of generators.
In Chapter 1 we give the basic background and notations. We also give a new characterization of the Conrad property for orderings. In Chapter 2, we use the new characterization of the Conradian property to give a classification of groups…
In this article, we study connections between representation theory and efficient solutions to the conjugacy problem on finitely generated groups. The main focus is on the conjugacy problem in conjugacy separable groups, where we measure…
We study systematically groups whose marked finite quotients form a recursive set. We give several definitions, and prove basic properties of this class of groups, and in particular emphasize the link between the growth of the depth…
A survey of problems, conjectures, and theorems about quasi-isometric classification and rigidity for finitely generated solvable groups.
We construct a finitely presented group with property (T) which can not act on on reasonable spaces. Such group is constructed using an generalization of Hall embedding theorem, where property (T) is added at the expense of weakening the…
It is known that in any free group the isolator of finitely generated subgroup is finitely generated subgroup. A very simple proof of this statement is proposed.
We study the space of ends of groups. For a finitely generated group, this is a Cantor space as soon as it is infinite. In contrast, we show that for infinitely generated countable groups, it exhibits several behaviors. For instance, we…
An isolated point on an algebraic curve is a closed point not belonging to a collection of points of the same degree parametrized by $\mathbf{P}^1$ or a positive rank abelian subvariety of the curve's Jacobian. We study the sets of…
We summarize several results about non-simplicity, solvability and normal structure of finite groups related to the number of conjugacy classes appearing in the product or the power of conjugacy classes. We also collect some problems that…
We study groups, exponential groups and ordered groups equipped with valuations. We investigate algebraic and topological features of such valued structures, and apply our findings in order to solve regular equations over groups using…
We give a self-contained introduction to isolated points on curves and their counterpoint, parameterized points, that situates these concepts within the study of the arithmetic of curves. In particular, we show how natural geometric…
Broadly speaking, a finiteness property of groups is any generalisation of the property of having finite order. A large part of infinite group theory is concerned with finiteness properties and the relationships between them. Profinite…
We study isolated points on the modular curves $X_{H}$, for $H$ a subgroup of $\operatorname{GL}_{2}(\mathbb{Z}/n \mathbb{Z})$ for some $n \geq 1$. In particular, we prove a single-sink theorem for such isolated points, which traces the…
We say that a subgroup $H$ is isolated in a group $G$ if for every $x\in G$ we have either $x\in H$ or $\langle x\rangle\cap H=1$. In this short note, we describe the set of isolated subgroups of a finite abelian group. The technique used…
Some boundedness properties of function spaces (considered as topological groups) are studied.
It is shown that there exists a finitely generated infinite simple group of infinite commutator width, and that the commutator width of a finitely generated infinite boundedly simple group can be arbitrarily large. Besides, such groups can…
We investigate indeterminate points in discrete integrable system. They appear in singularity confinement phenomenon naturally. We develop a method to analyse indeterminate points of dynamical maps and using this method we clarify behaviour…