Abstract
We provide the first examples of finitely generated simple groups that are amenable (and infinite). To this end, we prove that topological full groups of minimal systems are amenable. This follows from a general existence result on invariant states for piecewise-translations of the integers. The states are obtained by constructing a suitable family of densities on the classical Bernoulli space.
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[Bezuglyi-Medynets]
S. Bezuglyi and K. Medynets, "Full groups, flip conjugacy, and orbit equivalence of Cantor minimal systems," Colloq. Math., vol. 110, iss. 2, pp. 409-429, 2008.
@article {Bezuglyi-Medynets, MRKEY = {2353913},
AUTHOR = {Bezuglyi, S. and Medynets, K.},
TITLE = {Full groups, flip conjugacy, and orbit equivalence of {C}antor minimal systems},
JOURNAL = {Colloq. Math.},
FJOURNAL = {Colloquium Mathematicum},
VOLUME = {110},
YEAR = {2008},
NUMBER = {2},
PAGES = {409--429},
ISSN = {0010-1354},
CODEN = {CQMAAQ},
MRCLASS = {37B05 (20B99 28A60 28D05 37A20)},
MRNUMBER = {2353913},
MRREVIEWER = {Jan Kwiatkowski},
DOI = {10.4064/cm110-2-6},
ZBLNUMBER = {1142.37011},
} -
[Elek-Monod] G. Elek and N. Monod, On the topological full group of a minimal Cantor $\mathbf{Z}^2$-system.
@misc{Elek-Monod,
author={Elek, G. and Monod, N.},
TITLE={On the topological full group of a minimal {C}antor $\mathbf{Z}^2$-system},
NOTE={to appear in \emph{Proc. Amer. Math. Soc.}},
} -
[Grigorchuk-Medynets_v3] R. I. Grigorchuk and K. Medynets, Topological full groups are locally embeddable into finite groups.
@misc{Grigorchuk-Medynets_v3,
author={Grigorchuk, R. I. and Medynets, K.},
TITLE={Topological full groups are locally embeddable into finite groups},
ARXIV={1105.0719v3},
} -
[Glasner-Monod]
Y. Glasner and N. Monod, "Amenable actions, free products and a fixed point property," Bull. Lond. Math. Soc., vol. 39, iss. 1, pp. 138-150, 2007.
@article {Glasner-Monod, MRKEY = {2303529},
AUTHOR = {Glasner, Y. and Monod, N.},
TITLE = {Amenable actions, free products and a fixed point property},
JOURNAL = {Bull. Lond. Math. Soc.},
FJOURNAL = {Bulletin of the London Mathematical Society},
VOLUME = {39},
YEAR = {2007},
NUMBER = {1},
PAGES = {138--150},
ISSN = {0024-6093},
MRCLASS = {43A07 (20E06 20F65)},
MRNUMBER = {2303529},
MRREVIEWER = {Paul Jolissaint},
DOI = {10.1112/blms/bdl011},
ZBLNUMBER = {1207.43002},
} -
[Giordano-Putnam-Skau99]
T. Giordano, I. F. Putnam, and C. F. Skau, "Full groups of Cantor minimal systems," Israel J. Math., vol. 111, pp. 285-320, 1999.
@article {Giordano-Putnam-Skau99, MRKEY = {1710743},
AUTHOR = {Giordano, Thierry and Putnam, Ian F. and Skau, Christian F.},
TITLE = {Full groups of {C}antor minimal systems},
JOURNAL = {Israel J. Math.},
FJOURNAL = {Israel Journal of Mathematics},
VOLUME = {111},
YEAR = {1999},
PAGES = {285--320},
ISSN = {0021-2172},
CODEN = {ISJMAP},
MRCLASS = {46L55 (37B05 46L80 54H20)},
MRNUMBER = {1710743},
MRREVIEWER = {A. I. Danilenko},
DOI = {10.1007/BF02810689},
ZBLNUMBER = {0942.46040},
} -
[Juschenko-Salle] K. Juschenko and M. de la Salle, Invariant means of the wobbling group.
@misc{Juschenko-Salle,
author={Juschenko, K. and de~la Salle, M.},
TITLE={Invariant means of the wobbling group},
ARXIV= {1301.4736},
} -
[Matui06]
H. Matui, "Some remarks on topological full groups of Cantor minimal systems," Internat. J. Math., vol. 17, iss. 2, pp. 231-251, 2006.
@article {Matui06, MRKEY = {2205435},
AUTHOR = {Matui, Hiroki},
TITLE = {Some remarks on topological full groups of {C}antor minimal systems},
JOURNAL = {Internat. J. Math.},
FJOURNAL = {International Journal of Mathematics},
VOLUME = {17},
YEAR = {2006},
NUMBER = {2},
PAGES = {231--251},
ISSN = {0129-167X},
MRCLASS = {37B05 (37B10)},
MRNUMBER = {2205435},
MRREVIEWER = {Fred W. Roush},
DOI = {10.1142/S0129167X06003448},
ZBLNUMBER = {1109.37008},
} -
[Paterson] A. L. T. Paterson, Amenability, Providence, RI: Amer. Math. Soc., 1988, vol. 29.
@book {Paterson, MRKEY = {0961261},
AUTHOR = {Paterson, Alan L. T.},
TITLE = {Amenability},
SERIES = {Math. Surveys Monogr.},
VOLUME = {29},
PUBLISHER = {Amer. Math. Soc.},
ADDRESS = {Providence, RI},
YEAR = {1988},
PAGES = {xx+452},
ISBN = {0-8218-1529-6},
MRCLASS = {43-02 (22-02 43A07 46Lxx)},
MRNUMBER = {0961261},
MRREVIEWER = {C. Chou},
ZBLNUMBER = {0648.43001},
} -
[Putnam89]
I. F. Putnam, "The $C^*$-algebras associated with minimal homeomorphisms of the Cantor set," Pacific J. Math., vol. 136, iss. 2, pp. 329-353, 1989.
@article {Putnam89, MRKEY = {0978619},
AUTHOR = {Putnam, Ian F.},
TITLE = {The {$C\sp *$}-algebras associated with minimal homeomorphisms of the {C}antor set},
JOURNAL = {Pacific J. Math.},
FJOURNAL = {Pacific Journal of Mathematics},
VOLUME = {136},
YEAR = {1989},
NUMBER = {2},
PAGES = {329--353},
ISSN = {0030-8730},
CODEN = {PJMAAI},
MRCLASS = {46L80 (19K99 46L55 58F11)},
MRNUMBER = {0978619},
MRREVIEWER = {Klaus Thomsen},
DOI = {10.2140/pjm.1989.136.329},
ZBLNUMBER = {0631.46068},
} -
[vanDouwen]
E. K. van Douwen, "Measures invariant under actions of $F_2$," Topology Appl., vol. 34, iss. 1, pp. 53-68, 1990.
@article {vanDouwen, MRKEY = {1035460},
AUTHOR = {{van Douwen},
Eric K.},
TITLE = {Measures invariant under actions of {$F\sb 2$}},
JOURNAL = {Topology Appl.},
FJOURNAL = {Topology and its Applications},
VOLUME = {34},
YEAR = {1990},
NUMBER = {1},
PAGES = {53--68},
ISSN = {0166-8641},
CODEN = {TIAPD9},
MRCLASS = {43A05},
MRNUMBER = {1035460},
MRREVIEWER = {Joseph Max Rosenblatt},
DOI = {10.1016/0166-8641(90)90089-K},
ZBLNUMBER = {0701.43001},
}