Related papers: A short note about diffuse Bieberbach groups

It has been shown by several authors that there exists a non-solvable Bieberbach group of dimension $15$. In this note we show that this is in fact a minimal dimension for such kind of groups.

We determine which non-crystallographic, almost-crystallographic groups of dimension 4 have the $R_\infty$-property. We then calculate the Reidemeister spectra of the 3-dimensional almost-crystallographic groups and the 4-dimensional…

An $n$-dimensional closed flat manifold is said to be of diagonal type if the standard representation of its holonomy group $G$ is diagonal. An $n$-dimensional Bieberbach group of diagonal type is the fundamental group of such a manifold.…

We give a characterisation of Bieberbach manifolds which are geodesic boundaries of a compact flat manifold, and discuss the low dimensional cases, up to dimension 4.

An n-dimensional Bieberbach group is the fundamental group of a closed flat $n$-dimensional manifold. K. Dekimpe and P. Penninckx conjectured that an n-dimensional Bieberbach group can be generated by n elements. In this paper, we show that…

We discuss the notion of essential dimension of a finite group and explain its relation with birational algebraic geometry. We show how this leads to a (partial) classification of simple finite groups of essential dimension less than or…

Bowditch introduced the notion of diffuse groups as a geometric variation of the unique product property. We elaborate on various examples and non-examples, keeping the geometric point of view from Bowditch's paper. In particular, we…

We approach the quasi-isometric classification questions on Lie groups by considering low dimensional cases and isometries alongside quasi-isometries. First, we present some new results related to quasi-isometries between Heintze groups.…

We establish a lower bound for the representation dimension of all the classical Hecke algebras of types A, B and D. For all the type A algebras, and most of the algebras of types B and D, we also establish upper bounds. Moreover, we…

Leibniz algebras ${\mathcal E}_n$ were introduced as algebraic structure underlying U-duality. Algebras ${\mathcal E}_3$ derived from Bianchi three-dimensional Lie algebras are classified here. Two types of algebras are obtained:…

The article presents the structure of the automorphism groups of two types of non-nilpotent Leibniz algebras with a dimension of 3.

We classify the irreducible projective representations of symmetric and alternating groups of minimal possible and second minimal possible dimensions, and get a lower bound for the third minimal dimension. On the way we obtain some new…

We present a classification, up to isomorphisms, of all the homogeneous spaces of the Lorentz group with dimension lower than six. At the same time, we classify, up to conjugation, all the non-discrete closed subgroup of the Lorentz group…

In the article \cite{BM1.22}, the minimum volume entropy is introduced for each group of finite presentation and there we study some of its general properties. Another concept of minimum volume entropy for geometrically finite groups has…

In this paper we prove that for each dimension $n$ there are only finitely many isomorphism classes of pairs of groups $(\Gamma,\mathrm{N})$ such that $\Gamma$ is an $n$-dimensional crystallographic group and $\mathrm{N}$ is a normal…

We present about twenty conjectures, problems and questions about flat manifolds. Many of them build the bridges between the flat world and representation theory of the finite groups, hyperbolic geometry and dynamical systems.

We introduce the notion of the depth of a finite group $G$, defined as the minimal length of an unrefinable chain of subgroups from $G$ to the trivial subgroup. In this paper we investigate the depth of (non-abelian) finite simple groups.…

This is a survey paper about Ornstein-Uhlenbeck semigroups in infinite dimension, and their generators. We start from the classical Ornstein-Uhlenbeck semigroup in Wiener spaces and then discuss the general case in Hilbert spaces. Finally,…

The category of linear algebraic groups admits non-surjective epimorphisms. For simple algebraic groups of rank $2$ defined over algebraically closed fields, we show that the minimal dimension of a closed epimorphic subgroup is $3$.

Each Abelian subgroup of the fundamental group of a compact and locally simply connected $d$-dimensional length space with no conjugate points is isomorphic to $\mathbb{Z}^k$ for some $0 \leq k \leq d$. It follows from this and previously…