"Münchhausen trick" and amenability of self-similar groups

International Journal of Algebra and Computation

Volumn 15, Issue 5-6, 2005, Pages 907-937

  Kaimanovich, Vadim A ^a,b  

^a UNIVERSITÉ DE RENNES 1   (France)

^b JACOBS UNIVERSITY BREMEN   (Germany)

Author keywords

Amenability; Asymptotic entropy; Internal degrees of freedom; Liouville property; Random walk; Self similar group

Indexed keywords

EID: 28444440402     PISSN: 02181967     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218196705002694     Document Type: Conference Paper

References (21)

1. L. Bartholdi, Amenability of groups acting on trees (2002), arXiv: math.GR/0204076.
2. 2 - 1 (2002), arXiv: math.GR/0203244.
3. L. Bartholdi, R. Grigorchuk and V. Nekrashevych, From fractal groups to fractal sets (2002), arXiv: math.GR/0202001.
4. L. Bartholdi, R. I. Grigorchuk and Z. Šuniḱ, Branch Groups, in Handbook of Algebra, Vol. 3 (North-Holland, Amsterdam, 2003), pp. 989-1112. MR 2 035 113.
5. B. Bielefeld (ed.), Conformal Dynamics Problem List, Institute for Mathematical Sciences preprint series, No. 90-1 (SUNY Stony Brook, 1990), arXiv: math.DS/9201271.
6. L. Bartholdi and B. Virág, Amenability via random walks (2003), arXiv: math.GR/0305262.
7. A. Connes and E. J. Woods, Hyperfinite von Neumann algebras and Poisson boundaries of time dependent random walks, Pacific J. Math. 137(2) (1989) 225-243, MR 90h:46100.
8. A. Erschler, Boundary behaviour for groups of subexponential growth, Ann. Math (2) 160(3) (2004) 1183-1210.
9. A. Erschler, Liouville property for groups and manifolds, Invent. Math. 155(1) (2004) 55-80, MR 2 025 301.
10. Y. Glasner and S. Mozes, Automata and square complexes, preprint (2003).
11. R. I. Grigorchuk and A. Zuk, The lamplighter group as a group generated by a 2-state automaton, and its spectrum, Geom. Dedicata 87(1-9) (2001) 209-244, MR. 2002j:60009.
12. R. I. Grigorchuk and A. Zuk, On a torsion-free weakly branch group defined by a three state automaton, Int. J. Algebra Comput. 12(1-2) (2002) 223-246, International Conference on Geometric and Combinatorial Methods in Group Theory and Semigroup Theory (Lincoln, NE, 2000), MR 2003c: 20048.
13. R. I. Grigorchuk and A. Zuk, Spectral properties of a torsion-free weakly branch group defined by a three state automaton, Computational and Statistical Group Theory (Las Vegas, NV/Hoboken, NJ, 2001), Contemp. Math., Vol. 298 (Amer. Math. Soc., Providence, RI, 2002), pp. 57-82, MR 2003h:60011.
14. V. A. Kaimanovich, Brownian motion and harmonic functions on covering manifolds. An entropic approach, Soviet Math. Dokl. 33(3) (1986) 812-816, English translation, MR 88k:58163.
15. V. A. Kaimanovich, The Poisson boundary of covering Markov operators, Israel J. Math. 89(1-3) (1995) 77-134, MR 96k:60194.
16. d Actions (Warwick, 1993-1994), London Math. Soc. Lecture Note Series, Vol. 228 (Cambridge University Press, Cambridge, 1996), pp. 127-176, MR 97j:31008.
17. A. Krámli and D. Szász, Random walks with internal degrees of freedom. I. Local limit theorems, Z. Wahrsch. Verw. Gebiete 63(1) (1983) 85-95, MR 85f:60098.
18. V. A. Kaimanovich and A. M. Vershik, Random walks on discrete groups: boundary and entropy, Ann. Probab. 11(3) (1983) 457-490, MR 85d:60024.
19. V. Nekrashevych, Iterated monodromy groups (2003), arXiv: math.DS/0312306.
20. N. Th. Varopoulos, Long range estimates for Markov chains, Bull. Sci. Math. (2) 109(3) (1985) 225-252, MR 87j:60100.
21. B. Virág, A self-similar walk on a self-similar group, preprint (2003).