On geometric convergence of discrete groups

Shihai Yang

Displaying similar documents to “On geometric convergence of discrete groups”

Cellularity of a space of subgroups of a discrete group

Arkady G. Leiderman, Igor V. Protasov (2008)

Commentationes Mathematicae Universitatis Carolinae

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Given a discrete group $G$ , we consider the set $ℒ (G)$ of all subgroups of $G$ endowed with topology of pointwise convergence arising from the standard embedding of $ℒ (G)$ into the Cantor cube ${0, 1}^{G}$ . We show that the cellularity $c (ℒ (G)) \leq ℵ_{0}$ for every abelian group $G$ , and, for every infinite cardinal $τ$ , we construct a group $G$ with $c (ℒ (G)) = τ$ .

Pure subgroups

Ladislav Bican (2001)

Mathematica Bohemica

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Let $λ$ be an infinite cardinal. Set $λ_{0} = λ$ , define $λ_{i + 1} = 2^{λ_{i}}$ for every $i = 0, 1, \dots$ , take $μ$ as the first cardinal with $λ_{i} < μ$ , $i = 0, 1, \dots$ and put $κ = {(μ^{ℵ_{0}})}^{+}$ . If $F$ is a torsion-free group of cardinality at least $κ$ and $K$ is its subgroup such that $F / K$ is torsion and $| F / K | \leq λ$ , then $K$ contains a non-zero subgroup pure in $F$ . This generalizes the result from a previous paper dealing with $F / K$ $p$ -primary.

The dual group of a dense subgroup

William Wistar Comfort, S. U. Raczkowski, F. Javier Trigos-Arrieta (2004)

Czechoslovak Mathematical Journal

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Throughout this abstract, $G$ is a topological Abelian group and $\hat{G}$ is the space of continuous homomorphisms from $G$ into the circle group $𝕋$ in the compact-open topology. A dense subgroup $D$ of $G$ is said to determine $G$ if the (necessarily continuous) surjective isomorphism $\hat{G} ↠ \hat{D}$ given by $h \mapsto h | D$ is a homeomorphism, and $G$ is determined if each dense subgroup of $G$ determines $G$ . The principal result in this area, obtained independently by L. Außenhofer and M. J. Chasco, is the following: Every metrizable...