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Locally compact groups, residual Lie groups, and varieties generated by Lie groups. (English) Zbl 0858.22005

In what ways might a group \(G\) be “approximated by Lie groups”? Offering suitable citations to the literature, the authors discuss several properties of this sort, including these: (a) every neighborhood of \(1_G\) contains a closed [or even: compact] normal subgroup \(N\) of \(G\) such that \(G/N\) is Lie; (b) \(G\) is residual Lie: the class of continuous homomorphisms from \(G\) to Lie groups separates points; (c) \(G\) belongs to the variety generated by the class \(\mathbb{L}\) of Lie groups – that is, the smallest class containing \(\mathbb{L}\) and closed under passage to subgroups, Hausdorff quotients, and arbitrary products. A sequence of examples, locally compact when possible, distinguishes between the many classes discussed.
Among the paper’s new results are the construction, for locally compact residual Lie groups \(G = \langle G,{\mathcal T}\rangle\), of some coarser and “better behaved” topologies \(\mathcal U\) for which the identity function \(\text{id}:\langle G,{\mathcal T}\rangle \twoheadrightarrow \langle G,{\mathcal U}\rangle\) is an equidimensional immersion in this sense: for every one-parameter subgroup \(X:\mathbb{R}\to \langle G,{\mathcal U}\rangle\) there is a one-parameter subgroup \(X':\mathbb{R}\to\langle G,{\mathcal T}\rangle\) such that \(X =\text{id}\circ X'\). (For example, one may take for \(\mathcal U\) the group topology generated by sets of the form \(UN\) with \(U\) a \(\mathcal T\)-neighborhood of \(1_G\) and \(N\) a \(\mathcal T\)-compact, normal subgroup of \(G\); in this case \(\langle G,{\mathcal U}\rangle\) is also residual Lie.) Such a construction furnishes the result that for every locally compact residual Lie group \(G\) the component factor group \(G/G_0\) is residually discrete.

MSC:

22D05 General properties and structure of locally compact groups
20E26 Residual properties and generalizations; residually finite groups
14L10 Group varieties
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