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0

Let us call a topological group $\Gamma$ Muranov if for any compact sets $A,B\subset\Gamma$ and points $a,b,t$ the inclusions $aA\cup bB\subset A\cup B$ and $aA\cap bB\subset t(A\cap B)$ imply the equalities $aA\cup bB=A\cup B$ and $aA\cap bB=t(A\cap B)$. Theorem. A topological group $\Gamma$ is Muranov if each 3-generated subgroup of $\Gamma$ is discrete. ...

0

The answer to this question is affirmative up to a set of measure zero. Namely, for sets $A,B\subset\Gamma$ and points $a,b,t_1,t_2\in\Gamma$ with $aA\cup bA\subset t_1(A\cup B)$ and $aA\cap bB\subset t_2(A\cap B)$ one can show that $\lambda(aA\cup bB)=\lambda(t_1(A\cup B))$ and $\lambda(aA\cap bB)=\lambda(t_1(A\cap B))$ for every $\{a,b,t_1,t_2\}$-invariant ...

2

The answer to both problems (1 and 2) is negative: the Polish group $G=\mathbb Z^\omega$ contains a dense meager Borel subgroup $H$ (which can be written as the difference $H=A\setminus B$ of two $F_\sigma$-sets in $G$) which cannot be covered by countably many closed Haar-meager subsets of $G$. Such subgroup $H$ is constructed in Example 6.3 of this paper. ...

1

Every compactly generated locally compact group is either of polynomial growth, or it has a quotient that is just non-(polynomial growth). The same also works for a number of similar properties in place of 'polynomial growth', e.g. 'compact-by-nilpotent'. See Theorem 3.12 of this preprint: http://arxiv.org/abs/1509.06593 (it should say "for any compactly ...

2

This problem is considered in the recent papers by Hofmann, Kramer (http://arxiv.org/pdf/1301.5114.pdf) and Antonyan, Dobrowolski [Locally contractible coset spaces, Forum Mathematicum. 27:4 (2015), 2157–2175]. According to these papers, for a locally compact group $G$ and a closed subgroup $H$ in $G$ the homogeneous space $X=G/H$ is an Euclidean manifold if ...

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It seems that the conjecture (H2) can be confirmed with help of the recent result of Hofmann and Kramer (http://arxiv.org/pdf/1301.5114.pdf) who proved that for a compact topological group $G$ and a closed subgroup $H\subset G$ the homogeneous space $X=G/H$ is a manifold if and only if $X$ contains a non-empty open set contractible in $X$. This result of ...

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