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A proof of the Hilbert-Smith conjecture for actions by Lipschitz maps. (English) Zbl 0879.57025
The Hilbert-Smith conjecture asserts that among locally compact groups only Lie groups can act effectively on manifolds. It was proved to be equivalent to its special case when the acting group is the group of \(p\)-adic integers. S. Bochner and D. Montgomery proved this conjecture for groups acting on a manifold by diffeomorphisms [Ann. Math., II. Ser. 47, 639-653 (1946; Zbl 0061.04407)]. A simpler geometrical proof was obtained in 1991 by D. Repovš, A. B. Skopenkov and E. V. Ščepin [Proc. Am. Math. Soc. 124, No. 4, 1219-1226 (1996; Zbl 0863.53004)].
The present paper makes further progress on this problem by presenting a proof of the Lipschitz version of the Hilbert-Smith conjecture, i.e., it is proved that the group of the \(p\)-adic integers (\(p\) any prime) cannot act effectively by Lipschitz maps on any closed manifold.

MSC:
57S25 Groups acting on specific manifolds
54F65 Topological characterizations of particular spaces
26A24 Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems
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