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Actions of semisimple groups and discrete subgroups. (English) Zbl 0671.57028
Proc. Int. Congr. Math., Berkeley/Calif. 1986, Vol. 2, 1247-1258 (1987).
[For the entire collection see Zbl 0657.00005.]
Let \(V=\{primes\) in \({\mathbb{Z}}\}\cup \{\infty \}\). \({\mathbb{Q}}_ p\) denotes the p-adic numbers for p a finite prime and \({\mathbb{R}}\) for \(p=\infty\). For \(p\in V\), let \(G_ p\) be a connected semisimple algebraic group, \(G_ p({\mathbb{Q}}_ p)\) the group of \({\mathbb{Q}}_ p\)-points. Put \(G=\prod_{p\in S}G_ p({\mathbb{Q}}_ p)\), where \(S\subset V\) is a fixed finite subset. Let \(\Gamma\) \(\subset G\) be a lattice subgroup. In this report, the author discusses a much more recent program of understanding the realizations of G and \(\Gamma\) in another natural class of groups, namely, smooth transformation groups on compact manifolds. The results described are mainly due to the author.
Reviewer: J.Kubarski

57S25 Groups acting on specific manifolds
57S17 Finite transformation groups
22E40 Discrete subgroups of Lie groups
58A99 General theory of differentiable manifolds
22E50 Representations of Lie and linear algebraic groups over local fields