"MÜNCHHAUSEN TRICK" AND AMENABILITY OF SELF-SIMILAR GROUPS
2005 ◽
Vol 15
(05n06)
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pp. 907-937
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Keyword(s):
Group A
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The structure of a self-similar group G naturally gives rise to a transformation which assigns to any probability measure μ on G and any vertex w in the action tree of the group a new probability measure μw. If the measure μ is self-similar in the sense that μw is a convex combination of μ and the δ-measure at the group identity, then the asymptotic entropy of the random walk (G, μ) vanishes; therefore, the random walk is Liouville and the group G is amenable. We construct self-similar measures on several classes of self-similar groups, including the Grigorchuk group of intermediate growth.
2012 ◽
Vol 33
(7)
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pp. 1408-1421
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2015 ◽
Vol 25
(04)
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pp. 633-668
Keyword(s):
2013 ◽
Vol 34
(3)
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pp. 837-853
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Keyword(s):
Keyword(s):
2003 ◽
Vol 356
(1)
◽
pp. 393-414
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2016 ◽
Vol 37
(5)
◽
pp. 1480-1491
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Keyword(s):