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The Hilbert-Smith conjecture for Hölder actions. (English. Russian original) Zbl 0988.57021
Russ. Math. Surv. 52, No. 2, 407-408 (1997); translation from Usp. Mat. Nauk 52, No. 2, 173-174 (1997).
Introduction: The Hilbert-Smith conjecture asserts that every compact group effectively acting on a manifold is a Lie group. It is known that for the solution of this conjecture it is sufficient to prove that the group of $$p$$-adic numbers cannot act effectively on a manifold. It was shown in [D. Repovš and E. Shchepin, Math. Ann. 308, No. 2, 361-364 (1997; Zbl 0879.57025)] that the $$p$$-adic numbers cannot act on a manifold by Lipschitz homeomorphisms. In this note it is shown that the same is true for Hölder actions with a large Hölder index.

##### MSC:
 57S10 Compact groups of homeomorphisms 54H15 Transformation groups and semigroups (topological aspects) 22C05 Compact groups
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