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8.61 (1982)

Solved

Suppose that a locally compact group $G$ contains a subgroup that is topologically isomorphic to the additive group of the field of real numbers with natural topology. Is the space of all closed subgroups of $G$ connected in the Chabauty topology?

Contributor: I. V. Protasov

Progress

No, not always: let $H$ be the group of matrices of the form $\begin{pmatrix} 1 & z_1 & r \\ 0 & 1 & z_2 \\ 0 & 0 & 1 \end{pmatrix}$, where $z_1, z_2 \in \mathbb{Z}$ and $r \in \mathbb{R}$. Then $H$ is locally compact in the natural topology and contains a central subgroup topologically isomorphic to $\mathbb{R}$, but $L(H)$ is not connected (Yu. V. Tsybenko, Abstracts of 17th All-USSR Algebraic Conf., Part 1, Minsk, 1983, 213 (Russian)).

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