Timeline for When can a locally compact group be approximated by discrete subgroups?
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12 events
| when toggle format | what | by | license | comment | |
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| Jan 7, 2016 at 4:22 | comment | added | Colin Reid | I don't know if this helps for the application you have in mind, but for t.d.l.c. groups, an alternative way to approximate the group by discrete objects is to look instead at coset spaces $G/U$ where $U$ ranges over a base of identity neighbourhoods consisting of compact open subgroups. If $G$ is a SIN group, then you can also make $U$ normal in $G$, so $G/U$ is a discrete quotient group of $G$ in the natural sense. | |
| Jan 7, 2016 at 1:14 | comment | added | YCor | Btw I should add the usual definition of approximable by discrete subgroups (for a 2nd countable locally compact group $G$): the existence of a sequence $(H_n)$ of discrete subgroups such that for every nonempty open subset $U$ of $G$ there exists $N=N_U$ such that $H_n\cap U\neq\emptyset$ for all $n\ge N$ (if $(H_n)$ is ascending this just means that the union is dense). The above characterizations anyway are correct with either definition. | |
| Jan 7, 2016 at 1:10 | comment | added | YCor | At the opposite I'm not sure the case of totally disconnected locally compact groups is so well understood. | |
| Jan 7, 2016 at 1:10 | comment | added | YCor | By Kuranishi (1951), a simply connected nilpotent Lie group is approximable by discrete subgroups iff it has a cocompact lattice, iff its Lie algebra has a rational structure. He also proved that a nilpotent Lie group has is approximable by discrete subgroup iff it has a cocompact lattice, iff its Lie algebra has a rational structure such that the kernel of the universal covering is a rational subgroup. | |
| Jan 7, 2016 at 0:34 | comment | added | paul garrett | So now that @YCor's scholarship has "grounded" the question... we can hopefully-usefully restrict the question to a regime where the outcome is positive for the immediate purposes? Clarification? | |
| Jan 6, 2016 at 23:52 | comment | added | YCor | Second, the bare existence of a discrete cocompact subgroup is a quite strong property, which for instance implies that the group is unimodular. | |
| Jan 6, 2016 at 23:51 | comment | added | YCor | It is an old result of H. Toyama that if a connected Lie group is approximable by discrete subgroups, then it is nilpotent (and this even holds assuming only that the subgroups are discrete, not necessarily cocompact). By the way your definition should at least be called "approximable by discrete cocompact subgroups". | |
| Jan 6, 2016 at 23:48 | comment | added | YCor | A countable (= at most countable) locally compact group is discrete. So the weird "locally compact subgroup" just means discrete subgroup, and a discrete subgroup $H$ such that $G/H$ is compact is known as a "cocompact lattice" or "uniform lattice". | |
| Jan 6, 2016 at 23:40 | comment | added | Yemon Choi | Do you really need $G/G_n$ to be a group, or is it just compactness that you want? | |
| Jan 6, 2016 at 23:37 | comment | added | paul garrett | I fear that this property (strictly required) is difficult to achieve... My guess would be that, among real Lie groups, for example, only nilpotent groups would have this property. But this is just a superficial reaction... | |
| Jan 6, 2016 at 23:37 | comment | added | Yemon Choi | Are you assuming in your definition that each $G_n$ is closed in $G$? Without this assumption, the quotient topology on $G/G_n$ won't be Hausdorff, I think | |
| Jan 6, 2016 at 23:28 | history | asked | Jason Rute | CC BY-SA 3.0 |