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2021 ◽  
Vol 127 (3) ◽  
Author(s):  
Venuste Nyagahakwa ◽  
Gratien Haguma

In this paper, we prove that each topological group isomorphism of the additive topological group $(\mathbb{R},+)$ of real numbers onto itself preserves the non-Lebesgue measurability of Vitali selectors of $\mathbb{R}$. Inspired by Kharazishvili's results, we further prove that each finite union of Vitali selectors related to different countable dense subgroups of $(\mathbb{R}, +)$, is not measurable in the Lebesgue sense. From here, we produce a semigroup of sets, for which elements are not measurable in the Lebesgue sense. We finally show that the produced semigroup is invariant under the action of the group of all affine transformations of $\mathbb{R}$ onto itself.



2020 ◽  
Vol 275 ◽  
pp. 107000 ◽  
Author(s):  
Vitalij A. Chatyrko ◽  
Dmitri B. Shakhmatov


2020 ◽  
Vol 545 ◽  
pp. 159-173
Author(s):  
Alla S. Detinko ◽  
Willem A. de Graaf








2018 ◽  
Vol 61 (03) ◽  
pp. 523-533 ◽  
Author(s):  
KRISHNENDU GONGOPADHYAY ◽  
ABHISHEK MUKHERJEE ◽  
SUJIT KUMAR SARDAR

AbstractLet ℍ be the division ring of real quaternions. Let SL(2, ℍ) be the group of 2 × 2 quaternionic matrices $A={\scriptsize{(\begin{array}{l@{\quad}l} a & b \\ c & d \end{array})}}$ with quaternionic determinant det A = |ad − aca−1b| = 1. This group acts by the orientation-preserving isometries of the five-dimensional real hyperbolic space. We obtain discreteness criteria for Zariski-dense subgroups of SL(2, ℍ).



2018 ◽  
Vol 29 (3) ◽  
pp. 296-305 ◽  
Author(s):  
A. S. Detinko ◽  
D. L. Flannery ◽  
A. Hulpke
Keyword(s):  


2017 ◽  
Vol 09 (01) ◽  
pp. 27-49
Author(s):  
P. de la Harpe ◽  
D. Kotschick

In various classes of infinite groups, we identify groups that are presentable by products, i.e. groups having finite index subgroups which are quotients of products of two commuting infinite subgroups. The classes we discuss here include groups of small virtual cohomological dimension and irreducible Zariski dense subgroups of appropriate algebraic groups. This leads to applications to groups of positive deficiency, to fundamental groups of three-manifolds and to Coxeter groups. For finitely generated groups presentable by products we discuss the problem of whether the factors in a presentation by products may be chosen to be finitely generated.



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