Embedding free amalgams of discrete groups in non-discrete topological groups.

*(English)*Zbl 0974.22001
Cossey, John (ed.) et al., Geometric group theory down under. Proceedings of a special year in geometric group theory, Canberra, Australia, July 14-19, 1996. Berlin: de Gruyter. 203-223 (1999).

The theory of locally compact groups, especially compact groups, is very developed now. However outside the class of locally compact groups, the standard techniques do not apply. The method of graded diagrams and central extensions of diagrammatically aspherical groups, which has been introduced and developed by A. Yu. Ol’shanskii, seems to be very useful in understanding the structure of topological groups. The method of Ol’shanskii was extended by the second author to diagrams over free products and applied to factor groups of free products. The first step in introducing this technique to topological groups was a previous paper of the authors [Topology Appl. 84, 105-120 (1998; Zbl 0933.22003)], where any set of discrete groups without involutions was embedded in a non-discrete Hausdorff group with many discrete subgroups. The main result of that paper was given there without proof. In this paper the authors do not only provide the proof, but also give a generalization of the main result of that paper [the authors, op. cit.] The free amalgam \(\Omega^1\) of an arbitrary set of groups \(\{G_{\mu} \mid \mu \in I\}\) is defined to be the set \(\bigcup_{\mu \in I} G_{\mu}\) with \(G_{\mu} \cap G_{\nu} = \langle 1\rangle\) whenever \(\mu \not= \nu\). The mapping \(g: \Omega^1 \rightarrow G\) is a topological embedding of the free amalgam \(\Omega^1\) of a set of topological groups \(G_{\mu}\), \(\mu \in I\), into a topological group \(G\) if it is injective and \(G_{\mu}\) is topologically isomorphic to \(g(G_{\mu})\) for each \(\mu \in I\). The main result of this paper is the following theorem.

Theorem 1.3. Assume that the set of non-identity groups \(\{G_{\mu}\mid \mu \in I\}\) contains either three groups or two groups one of which has order at least 3. Then the free amalgam \({\Omega^1}\) of the groups \(\{G_{\mu}\mid \mu \in I\}\) can be embedded in a group \(G = gp{\Omega^1}\) in such a way that, for each fixed cardinal \(\beta < |G|\), \(G\) admits a non-discrete Hausdorff topology such that (1) \(G\) is a 0-dimensional topological group; (2) every neighbourhood is of cardinality \(|G|\); (3) if a subgroup \(M\) of \(G\) is conjugate to a subgroup \(G_{\mu}\) for some \(\mu \in I\) or \(M\) is a finite extension of a cyclic group, then \(M\) is discrete; (4) every subgroup of \(G\) of cardinality \(\gamma \leq \beta\) is discrete; (5) if \(\beta\) is finite, then \(G\) may be chosen to be metrizable.

For the entire collection see [Zbl 0910.00040].

Theorem 1.3. Assume that the set of non-identity groups \(\{G_{\mu}\mid \mu \in I\}\) contains either three groups or two groups one of which has order at least 3. Then the free amalgam \({\Omega^1}\) of the groups \(\{G_{\mu}\mid \mu \in I\}\) can be embedded in a group \(G = gp{\Omega^1}\) in such a way that, for each fixed cardinal \(\beta < |G|\), \(G\) admits a non-discrete Hausdorff topology such that (1) \(G\) is a 0-dimensional topological group; (2) every neighbourhood is of cardinality \(|G|\); (3) if a subgroup \(M\) of \(G\) is conjugate to a subgroup \(G_{\mu}\) for some \(\mu \in I\) or \(M\) is a finite extension of a cyclic group, then \(M\) is discrete; (4) every subgroup of \(G\) of cardinality \(\gamma \leq \beta\) is discrete; (5) if \(\beta\) is finite, then \(G\) may be chosen to be metrizable.

For the entire collection see [Zbl 0910.00040].

Reviewer: Leonid Kurdachenko (Dnepropetrovsk)

##### MSC:

22A05 | Structure of general topological groups |