Abstract
For a countable amenable group $\Gamma$ and an element $f$ in the integral group ring $\mathbb{Z}\Gamma$ being invertible in the group von Neumann algebra of $\Gamma$, we show that the entropy of the shift action of $\Gamma$ on the Pontryagin dual of the quotient of $\mathbb{Z}\Gamma$ by its left ideal generated by $f$ is the logarithm of the Fuglede-Kadison determinant of $f$. For the proof, we establish an $\ell^p$-version of Rufus Bowen’s definition of topological entropy, addition formulas for group extensions of countable amenable group actions, and an approximation formula for the Fuglede-Kadison determinant of $f$ in terms of the determinants of perturbations of the compressions of $f$.
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CODEN = {ANMAAH},
MRCLASS = {37A15 (37A25 37A35)},
MRNUMBER = {1779565},
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TITLE = {Erratum: ``{I}ntegrality of {$L\sp 2$}-{B}etti numbers''},
JOURNAL = {Math. Ann.},
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VOLUME = {322},
YEAR = {2002},
NUMBER = {2},
PAGES = {421--422},
ISSN = {0025-5831},
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MRCLASS = {55N25 (16S34 46L99 58J22)},
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DOI = {10.1007/s002080100282},
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AUTHOR = {Schick, Thomas},
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JOURNAL = {Trans. Amer. Math. Soc.},
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NUMBER = {8},
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@book {Sch, MRKEY = {1345152},
AUTHOR = {Schmidt, Klaus},
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SERIES = {Progr. Math.},
VOLUME = {128},
PUBLISHER = {Birkhäuser},
ADDRESS = {Basel},
YEAR = {1995},
PAGES = {xviii+310},
ISBN = {3-7643-5174-8},
MRCLASS = {28D15 (11K55 22D40 28D20)},
MRNUMBER = {1345152},
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ZBLNUMBER = {0833.28001},
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@incollection {Schmidt01, MRKEY = {1905342},
AUTHOR = {Schmidt, Klaus},
TITLE = {The dynamics of algebraic {$\Bbb Z\sp d$}-actions},
BOOKTITLE = {European {C}ongress of {M}athematics, {V}ol. {I}},
VENUE={{B}arcelona, 2000},
SERIES = {Progr. Math.},
VOLUME = {201},
PAGES = {543--553},
PUBLISHER = {Birkhäuser},
ADDRESS = {Basel},
YEAR = {2001},
MRCLASS = {37A15 (28D20)},
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JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {111},
YEAR = {1993},
NUMBER = {1},
PAGES = {69--76},
ISSN = {0020-9910},
CODEN = {INVMBH},
MRCLASS = {22D40},
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ZBLNUMBER = {0824.28012},
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@book {Takesaki, MRKEY = {1873025},
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NUMBER = {3-4},
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@incollection {Weiss01, MRKEY = {1819193},
AUTHOR = {Weiss, Benjamin},
TITLE = {Monotileable amenable groups},
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SERIES = {Amer. Math. Soc. Transl. Ser. 2},
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@article {Yuz1, MRKEY = {0194588},
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TITLE = {Metric properties of the endomorphisms of compact groups},
JOURNAL = {Izv. Akad. Nauk SSSR Ser. Mat.},
FJOURNAL = {Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya},
VOLUME = {29},
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PAGES = {1295--1328},
ISSN = {0373-2436},
MRCLASS = {28.75 (10.33)},
NOTE={(Russian); translated in {\it Amer. Math. Soc. Transl.} {\bf 66} (1968), 63--98},
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@article{Yuz2,
author={Yuzvinski{\u\i},
S. A.},
TITLE={Computing the entropy of a group of endomorphisms},
JOURNAL={Sibirsk. Mat. \^{Z}.},
VOLUME={8},
YEAR={1967},
PAGES={230--239},
NOTE={(Russian); translated in {\it Siberian Math. J.} {\bf 8} (1967), 172--178},
ZBLNUMBER = {0211.34901},
}