In measure theoretic language there is a notion of matched pair of locally compact (l.c.) groups due to Baaj-Skandalis-Vaes. A pair $(G_{1}, G_{2})$ is called a matched pair of l.c. groups if there exists a l.c. group $G$ such that, $G_{1}$, $G_{2}$ are closed subgroups of $G$, intersection of $G_{1}$ and $G_{2}$ is trivial inside $G$ and the complement of $G_{1}G_{2}$ is of measure zero in $G$. Does there exists any notion which replace the last condition by some topological criterion?
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$\begingroup$ What do you mean by “topological”? Without using the group structure? Only regarding $G_1G_2$ as a subspace of $G$? $\endgroup$– The UserCommented Jul 20, 2013 at 19:23
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$\begingroup$ I think it means topological as opposed to measure-theoretic. $\endgroup$– MTSCommented Jul 20, 2013 at 19:24
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1$\begingroup$ @MTS “of measure zero with respect to a Haar measure” only depends on the topological-group-structure. It seems to be a plausible interpretation of “topological criterion” to take this as a valid answer. $\endgroup$– The UserCommented Jul 20, 2013 at 19:50
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$\begingroup$ @TheUser yes, of course you're right. $\endgroup$– MTSCommented Jul 20, 2013 at 19:51
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1$\begingroup$ You may want to look at the notion of quasi-product of Caprace-Monod: a topological group $G$ is the quasi-product of closed, normal subgroups $N_1,...,N_k$ if the multiplication map $N_1\times...\times N_k\rightarrow G$ is injective with dense image. Caprace and Monod construct examples of center-free quasi-products, some of them NOT being direct products. See Appendix II in arxiv.org/pdf/0811.4101.pdf $\endgroup$– Alain ValetteCommented Jul 21, 2013 at 16:03
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I suppose you could ask that the complement of $G_1G_2$ is nowhere dense, or more generally a meagre set. But whether this notion is appropriate or not really depends on what application you have in mind.
Also, unless I am missing something, isn't every pair of locally compact groups is a matched pair? Just take $G = G_1 \times G_2$.
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$\begingroup$ Thanks for your comment. Obviously direct product provides the trivial matching. But I think nowhere dense is a good candidate for that. $\endgroup$– SutanuCommented Jul 20, 2013 at 19:34
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$\begingroup$ Why isn’t it trivial in the context of Baaj-Skandalis-Vaes? $\endgroup$– The UserCommented Jul 20, 2013 at 19:51
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1$\begingroup$ The point is not that this notion admits trivial cases, but that it admits highly non trivial ones. The emphasis is not on whether a given pair is a matched pair but whether a group $G$ can be reconstructed from a matched pair... $\endgroup$ Commented Jul 21, 2013 at 9:02
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$\begingroup$ @NicolaCiccoli Yes, I wouldn't suspect Baaj-Skandalis-Vaes to come up with a trivial notion! It just seems a funny way to phrase the definition - it makes more sense if $G$ is the focus of the definition rather than $G_1$ and $G_2$. $\endgroup$– MTSCommented Jul 21, 2013 at 11:40
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$\begingroup$ Previously Baaj and Skandalis considered the map $(g_{1},g_{2})\to g_{1}g_{2}$ to be a homeomorphism. The Definition mentioned above was one step generalization. As per my understanding the goal was is to carry forward Drinfeld's bicrossed product construction in operator algebraic framework. General theory of Double crossed product for quantum groups in operator algebraic setting is due to Baaj-Vaes. One of the delicate parts of their construction was to give an explicit description of the $C^{*}$-algebra of the double crossed product. $\endgroup$– SutanuCommented Jul 21, 2013 at 12:17