Compact open subgroups in simple totally disconnected groups
So far, specific examples of totally disconnected, locally compact topological groups have been discussed in the literature which are simple or topologically simple (i.e., non-trivial and without non-trivial, proper, closed normal subgroups). The article under review can be seen as the starting point of the investigation of topologically simple, totally disconnected groups in general. It is known that all connected locally compact groups which are simple are real Lie groups. They are determined by their Lie algebra, and hence by their local structure. The article under review explores the impact of topological simplicity of a totally disconnected locally compact group \(G\) on its local structure. For compactly generated, non-discrete \(G\), it is shown that none of its compact open subgroups is solvable (Theorem 2.2) and that every compact open subgroup of \(G\) has an infinite \(p\)-Sylow subgroup for some prime \(p\) (cf. Corollary 2.9). Furthermore, \(G\) is metrizable (Proposition 2.1). The article also introduces a new class of topologically simple, totally disconnected groups which are not compactly generated. These are certain direct limits of profinite groups (with open bonding maps). The examples show that most of the above results become false for topologically simple groups which are not compactly generated: such groups need not be metrizable, and some of them have abelian open subgroups. The author also describes two topologically simple groups which are not isomorphic as topological groups, although they have isomorphic compact open subgroups. Again, these pathological examples are not compactly generated. Whether two compactly generated, topologically simple groups possessing isomorphic compact open subgroups have to be isomorphic is one of the challenging open problems formulated at the end of the article (the words ``compactly generated'' have to be added in Problem 4.3). It is to be expected that the questions raised will attract much more attention soon. Thomas Weigel already announced complementary results with a similar thrust, obtained using abstract commensurators of profinite groups.
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