Related papers: Uniformly perfect finitely generated simple left o…
A fundamental notion in group theory, which originates in an article of Ulam and von Neumann from $1947$ is uniform simplicity. A group $G$ is said to be $n$-uniformly simple for $n \in \mathbf{N}$ if for every $f,g\in G\setminus \{id\}$,…
The goal of this note is to provide yet another proof of the following theorem of Golod: there exists an infinite finitely generated group $G$ such that every element of $G$ has finite order. Our proof is based on the Nielsen-Schreier index…
It is shown that there exist finitely generated infinite simple groups of infinite commutator width and infinite square width on which there exists no stably unbounded conjugation-invariant norm, and in particular stable commutator length…
We prove that every finite simple group of Lie type $G$ can be generated by three regular unipotent elements. In certain cases we show that two regular unipotents are sufficient to generate $G$.
We study finite groups $G$ with elements $g$ such that $\lvert \mathbf{C}_G(g)\rvert = \lvert G:G' \rvert$. (Such elements generalize fixed-point-free automorphisms of finite groups.) We show that these groups have a unique conjugacy class…
We prove two results. (1) There is an absolute constant $D$ such that for any finite quasisimple group $S$, given 2D arbitrary automorphisms of $S$, every element of $S$ is equal to a product of $D$ `twisted commutators' defined by the…
A finite group $G$ is \emph{coprimely-invariably generated} if there exists a set of generators $\{g_1, ..., g_u\}$ of $G$ with the property that the orders $|g_1|, ..., |g_u|$ are pairwise coprime and that for all $x_1, ..., x_u \in G$ the…
In this paper we describe finitely generated groups $H$ universally equivalent (with constants from $G$ in the language) to a given torsion-free relatively hyperbolic group $G$ with free abelian parabolics. It turns out that, as in the free…
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…
A subset S of a group G invariably generates G if G = <s^(g(s)) | s in S> for each choice of g(s) in G, s in S. In this paper we study invariable generation of infinite groups, with emphasis on linear groups. Our main result shows that a…
We prove that if a finite group $G$ has a representation with fixity $f$, then it acts freely and homologically trivially on a finite CW-complex homotopy equivalent to a product of $f+1$ spheres. This shows, in particular, that every finite…
We prove that every countable left-ordered group embeds into a finitely generated left-ordered simple group. Moreover, if the first group has a computable left-order, then the simple group also has a computable left-order. We also obtain a…
Let F be a non-archimedean local field of characteristic zero whose residue field has at least three elements. Let G be an almost simple linear algebraic group over F, with rank_F(G) >= 2. Let X be a simply connected symmetric space of…
We prove the following result: Let $(X,g_0)$ be a complete, connected 4-manifold with uniformly positive isotropic curvature and with bounded geometry. Then there is a finite collection $\mathcal{F}$ of manifolds of the form $\mathbb{S}^3…
In this article we produce an example of a non-residually finite group which admits a uniformly proper action on a Gromov hyperbolic space.
We show that any free product of two countable groups, one of them being infinite, admits a faithful and homogeneous action on the Random Graph. We also show that a large class of HNN extensions or free products, amalgamated over a finite…
A regular left-order on finitely generated group $G$ is a total, left-multiplication invariant order on $G$ whose corresponding positive cone is the image of a regular language over the generating set of the group under the evaluation map.…
Uninorms play a prominent role both in the theory and the applications of Aggregations and Fuzzy Logic. In this paper the class of group-like uninorms is introduced and characterized. First, two variants of a general construction -- called…
A group $G$ is called automatically continuous if any homomorphism from a completely metrizable or locally compact Hausdorff group to $G$ has open kernel. In this paper, we study preservation of automatic continuity under group-theoretic…
We prove that if $X$ is a compact, oriented, connected $4$-dimensional smooth manifold, possibly with boundary, satisfying $\chi(X)\neq 0$, then there exists an integer $C\geq 1$ such that any finite group $G$ acting smoothly and…