Choquet-Deny groups and the infinite conjugacy class property

Joshua Frisch; Yair Hartman; Omer Tamuz; Pooya Vahidi Ferdowsi

doi:https://doi.org/10.4007/annals.2019.190.1.5

Abstract

A countable discrete group $G$ is called Choquet-Deny if for every non-degenerate probability measure $\mu$ on $G$, it holds that all bounded $\mu$-harmonic functions are constant. We show that a finitely generated group $G$ is Choquet-Deny if and only if it is virtually nilpotent. For general countable discrete groups, we show that $G$ is Choquet-Deny if and only if none of its quotients has the infinite conjugacy class property. Moreover, when $G$ is not Choquet-Deny, then this is witnessed by a symmetric, finite entropy, non-degenerate measure.

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[bader2006factor] $Go to document$ U. Bader and Y. Shalom, "Factor and normal subgroup theorems for lattices in products of groups," Invent. Math., vol. 163, iss. 2, pp. 415-454, 2006.

Show bibtex

@ARTICLE{bader2006factor, author = {Bader, Uri and Shalom, Yehuda}, title = {Factor and normal subgroup theorems for lattices in products of groups}, journal = {Invent. Math.}, fjournal = {Inventiones Mathematicae}, volume = {163}, year = {2006}, number = {2}, pages = {415--454}, issn = {0020-9910}, mrclass = {22E40 (28D15 43A05 43A07)}, mrnumber = {2207022}, mrreviewer = {Alain Valette}, doi = {10.1007/s00222-005-0469-5}, url = {https://doi.org/10.1007/s00222-005-0469-5}, zblnumber = {1085.22005}, }
[bartholdi2017poisson] $Go to document$ L. Bartholdi and A. Erschler, "Poisson-Furstenberg boundary and growth of groups," Probab. Theory Related Fields, vol. 168, iss. 1-2, pp. 347-372, 2017.

Show bibtex

@ARTICLE{bartholdi2017poisson, author = {Bartholdi, Laurent and Erschler, Anna}, title = {Poisson-{F}urstenberg boundary and growth of groups}, journal = {Probab. Theory Related Fields}, fjournal = {Probability Theory and Related Fields}, volume = {168}, year = {2017}, number = {1-2}, pages = {347--372}, issn = {0178-8051}, mrclass = {60B15 (20F65 20F69 43A07)}, mrnumber = {3651055}, mrreviewer = {Michael Voit}, doi = {10.1007/s00440-016-0712-6}, url = {https://doi.org/10.1007/s00440-016-0712-6}, zblnumber = {06734675}, }
[blackwell1955transient] $Go to document$ D. Blackwell, "On transient Markov processes with a countable number of states and stationary transition probabilities," Ann. Math. Statist., vol. 26, pp. 654-658, 1955.

Show bibtex

@ARTICLE{blackwell1955transient, author = {Blackwell, David}, title = {On transient {M}arkov processes with a countable number of states and stationary transition probabilities}, journal = {Ann. Math. Statist.}, fjournal = {Annals of Mathematical Statistics}, volume = {26}, year = {1955}, pages = {654--658}, issn = {0003-4851}, mrclass = {60.0X}, mrnumber = {0075479}, mrreviewer = {D. G. Kendall}, doi = {10.1214/aoms/1177728425}, url = {https://doi.org/10.1214/aoms/1177728425}, zblnumber = {0066.11303}, }
[cd1960] G. Choquet and J. Deny, "Sur l’équation de convolution $\mu =\mu \ast \sigma $," C. R. Acad. Sci. Paris, vol. 250, pp. 799-801, 1960.

Show bibtex

@ARTICLE{cd1960, author = {Choquet, Gustave and Deny, Jacques}, title = {Sur l'équation de convolution {$\mu =\mu \ast \sigma $}}, journal = {C. R. Acad. Sci. Paris}, volume = {250}, year = {1960}, pages = {799--801}, mrclass = {42.00}, mrnumber = {0119041}, mrreviewer = {E. Hewitt}, zblnumber = {0093.12802}, }
[duguid1956fc] $Go to document$ A. M. Duguid and D. H. McLain, "FC-nilpotent and FC-soluble groups," Proc. Cambridge Philos. Soc., vol. 52, pp. 391-398, 1956.

Show bibtex

@ARTICLE{duguid1956fc, author = {Duguid, A. M. and McLain, D. H.}, title = {F{C}-nilpotent and {FC}-soluble groups}, journal = {Proc. Cambridge Philos. Soc.}, volume = {52}, year = {1956}, pages = {391--398}, mrclass = {20.0X}, mrnumber = {0081280}, mrreviewer = {P. Hall}, zblnumber = {0071.02204}, doi = {10.1017/S030500410003139X}, }
[dynkin1961random] $Go to document$ E. B. Dynkin and M. B. Maljutov, "Random walk on groups with a finite number of generators," Dokl. Akad. Nauk SSSR, vol. 137, pp. 1042-1045, 1961.

Show bibtex

@ARTICLE{dynkin1961random, author = {Dynkin, E. B. and Maljutov, M. B.}, title = {Random walk on groups with a finite number of generators}, journal = {Dokl. Akad. Nauk SSSR}, fjournal = {Doklady Akademii Nauk SSSR}, volume = {137}, year = {1961}, pages = {1042--1045}, issn = {0002-3264}, mrclass = {60.08}, mrnumber = {0131904}, mrreviewer = {H. Kesten}, zblnumber = {0214.44101}, url = {http://mi.mathnet.ru/eng/dan/v137/i5/p1042}, }
[erschler2004boundary] $Go to document$ A. Erschler, "Boundary behavior for groups of subexponential growth," Ann. of Math. (2), vol. 160, iss. 3, pp. 1183-1210, 2004.

Show bibtex

@ARTICLE{erschler2004boundary, author = {Erschler, Anna}, title = {Boundary behavior for groups of subexponential growth}, journal = {Ann. of Math. (2)}, fjournal = {Annals of Mathematics. Second Series}, volume = {160}, year = {2004}, number = {3}, pages = {1183--1210}, issn = {0003-486X}, mrclass = {20F65 (20F05 20P05 60G50)}, mrnumber = {2144977}, mrreviewer = {Anders Karlsson}, doi = {10.4007/annals.2004.160.1183}, url = {https://doi.org/10.4007/annals.2004.160.1183}, zblnumber = {1089.20025}, }
[erschler2004liouville] $Go to document$ A. Erschler, "Liouville property for groups and manifolds," Invent. Math., vol. 155, iss. 1, pp. 55-80, 2004.

Show bibtex

@ARTICLE{erschler2004liouville, author = {Erschler, Anna}, title = {Liouville property for groups and manifolds}, journal = {Invent. Math.}, fjournal = {Inventiones Mathematicae}, volume = {155}, year = {2004}, number = {1}, pages = {55--80}, issn = {0020-9910}, mrclass = {43A85 (20P05 28D20 43A07 60B15 60G50 60J50)}, mrnumber = {2025301}, mrreviewer = {Anatoly N. Kochubei}, doi = {10.1007/s00222-003-0314-7}, url = {https://doi.org/10.1007/s00222-003-0314-7}, zblnumber = {1043.60006}, }
[frisch2018strong] J. Frisch, O. Tamuz, and P. V. Ferdowsi, Strong amenability and the infinite conjugacy class property, 2018.

Show bibtex

@MISC{frisch2018strong, author = {Frisch, J. and Tamuz, O. and Ferdowsi, P. V.}, title = {Strong amenability and the infinite conjugacy class property}, year = {2018}, arxiv = {1801.04024}, zblnumber = {}, }
[furman2002random] $Go to document$ A. Furman, "Random walks on groups and random transformations," in Handbook of Dynamical Systems, Vol. 1A, North-Holland, Amsterdam, 2002, pp. 931-1014.

Show bibtex

@INCOLLECTION{furman2002random, author = {Furman, Alex}, title = {Random walks on groups and random transformations}, booktitle = {Handbook of Dynamical Systems, {V}ol. 1{A}}, pages = {931--1014}, publisher = {North-Holland, Amsterdam}, year = {2002}, mrclass = {60G50 (37A50 37H15)}, mrnumber = {1928529}, mrreviewer = {Gernot Greschonig}, doi = {10.1016/S1874-575X(02)80014-5}, url = {https://doi.org/10.1016/S1874-575X(02)80014-5}, zblnumber = {1053.60045}, }
[furstenberg2009stationary] $Go to document$ H. Furstenberg and E. Glasner, "Stationary dynamical systems," in Dynamical Numbers—Interplay Between Dynamical Systems and Number Theory, Amer. Math. Soc., Providence, RI, 2010, vol. 532, pp. 1-28.

Show bibtex

@INCOLLECTION{furstenberg2009stationary, author = {Furstenberg, Hillel and Glasner, Eli}, title = {Stationary dynamical systems}, booktitle = {Dynamical Numbers---Interplay Between Dynamical Systems and Number Theory}, series = {Contemp. Math.}, volume = {532}, pages = {1--28}, publisher = {Amer. Math. Soc., Providence, RI}, year = {2010}, mrclass = {22D40 (28D05 37A15 37A30 37A40 37A50)}, mrnumber = {2762131}, mrreviewer = {Thomas Ward}, doi = {10.1090/conm/532/10481}, url = {https://doi.org/10.1090/conm/532/10481}, zblnumber = {1218.22004}, }
[furstenberg1963noncommuting] $Go to document$ H. Furstenberg, "Noncommuting random products," Trans. Amer. Math. Soc., vol. 108, pp. 377-428, 1963.

Show bibtex

@ARTICLE{furstenberg1963noncommuting, author = {Furstenberg, Harry}, title = {Noncommuting random products}, journal = {Trans. Amer. Math. Soc.}, fjournal = {Transactions of the American Mathematical Society}, volume = {108}, year = {1963}, pages = {377--428}, issn = {0002-9947}, mrclass = {60.08 (60.66)}, mrnumber = {0163345}, mrreviewer = {G.-C. Rota}, doi = {10.2307/1993589}, url = {https://doi.org/10.2307/1993589}, zblnumber = {0203.19102}, }
[furstenberg1971random] H. Furstenberg, "Random walks and discrete subgroups of Lie groups," in Advances in Probability and Related Topics, Vol. 1, Dekker, New York, 1971, pp. 1-63.

Show bibtex

@INCOLLECTION{furstenberg1971random, author = {Furstenberg, Harry}, title = {Random walks and discrete subgroups of {L}ie groups}, booktitle = {Advances in {P}robability and {R}elated {T}opics, {V}ol. 1}, pages = {1--63}, publisher = {Dekker, New York}, year = {1971}, mrclass = {28.75 (22.00)}, mrnumber = {0284569}, mrreviewer = {H. Kesten}, zblnumber = {0221.22008}, }
[furstenberg1973boundary] H. Furstenberg, "Boundary theory and stochastic processes on homogeneous spaces," in Harmonic Analysis on Homogeneous Spaces (Proc. Sympos. Pure Math., Vol. XXVI, Williams Coll., Williamstown, Mass., 1972), Amer. Math. Soc., Providence, R.I., 1973, pp. 193-229.

Show bibtex

@incollection{furstenberg1973boundary, author = {Furstenberg, Harry}, title = {Boundary theory and stochastic processes on homogeneous spaces}, booktitle = {Harmonic {A}nalysis on {H}omogeneous {S}paces ({P}roc. {S}ympos. {P}ure {M}ath., {V}ol. {XXVI}, {W}illiams {C}oll., {W}illiamstown, {M}ass., 1972)}, pages = {193--229}, publisher = {Amer. Math. Soc., Providence, R.I.}, year = {1973}, mrclass = {22E40 (60J50)}, mrnumber = {0352328}, mrreviewer = {R. G. Laha}, zblnumber = {0289.22011}, }
[glasner1976choquet] S. Glasner, "On Choquet-Deny measures," Ann. Inst. H. Poincaré Sect. B (N.S.), vol. 12, iss. 1, pp. 1-10, 1976.

Show bibtex

@ARTICLE{glasner1976choquet, author = {Glasner, Shmuel}, title = {On {C}hoquet-{D}eny measures}, journal = {Ann. Inst. H. Poincaré Sect. B (N.S.)}, volume = {12}, year = {1976}, number = {1}, pages = {1--10}, mrclass = {60J05 (31C05 43A05 60B15)}, mrnumber = {0488299}, mrreviewer = {A. Mukherjea}, zblnumber = {0349.60006}, }
[glasner1976proximal] $Go to document$ S. Glasner, Proximal Flows, Springer-Verlag, New York, 1976, vol. 517.

Show bibtex

@BOOK{glasner1976proximal, author = {Glasner, Shmuel}, title = {Proximal Flows}, series = {Lect. Notes in Math.}, volume = {517}, publisher = {Springer-Verlag, New York}, year = {1976}, pages = {viii+153}, mrclass = {54H20 (22E30)}, mrnumber = {0474243}, mrreviewer = {I. U. Bronstein}, zblnumber = {0322.54017}, doi = {10.1007/BFb0080139}, }
[guivarc1973croissance] $Go to document$ Y. Guivarc’h, "Croissance polynomiale et périodes des fonctions harmoniques," Bull. Soc. Math. France, vol. 101, pp. 333-379, 1973.

Show bibtex

@ARTICLE{guivarc1973croissance, author = {Guivarc'h, Yves}, title = {Croissance polynomiale et périodes des fonctions harmoniques}, journal = {Bull. Soc. Math. France}, fjournal = {Bulletin de la Société Mathématique de France}, volume = {101}, year = {1973}, pages = {333--379}, issn = {0037-9484}, mrclass = {22D05 (43A05 60B15)}, mrnumber = {0369608}, mrreviewer = {H. Heyer}, doi = {10.24033/bsmf.1764}, zblnumber = {0294.43003}, }
[jaworski2004countable] $Go to document$ W. Jaworski, "Countable amenable identity excluding groups," Canad. Math. Bull., vol. 47, iss. 2, pp. 215-228, 2004.

Show bibtex

@ARTICLE{jaworski2004countable, author = {Jaworski, Wojciech}, title = {Countable amenable identity excluding groups}, journal = {Canad. Math. Bull.}, fjournal = {Canadian Mathematical Bulletin. Bulletin Canadien de Mathématiques}, volume = {47}, year = {2004}, number = {2}, pages = {215--228}, issn = {0008-4395}, mrclass = {47A35 (22D10 43A05 43A07 60B15 60G50)}, mrnumber = {2059416}, mrreviewer = {C. R. E. Raja}, doi = {10.4153/CMB-2004-021-1}, url = {https://doi.org/10.4153/CMB-2004-021-1}, zblnumber = {1062.22010}, }
[jaworski2007choquet] $Go to document$ W. Jaworski and R. E. C. Raja, "The Choquet-Deny theorem and distal properties of totally disconnected locally compact groups of polynomial growth," New York J. Math., vol. 13, pp. 159-174, 2007.

Show bibtex

@ARTICLE{jaworski2007choquet, author = {Jaworski, Wojciech and Raja, C. Robinson Edward}, title = {The {C}hoquet-{D}eny theorem and distal properties of totally disconnected locally compact groups of polynomial growth}, journal = {New York J. Math.}, fjournal = {New York Journal of Mathematics}, volume = {13}, year = {2007}, pages = {159--174}, issn = {1076-9803}, mrclass = {60B15 (22D05 43A05 60J50)}, mrnumber = {2336237}, mrreviewer = {Wilfried Hazod}, url = {http://nyjm.albany.edu:8000/j/2007/13_159.html}, zblnumber = {1118.60008}, }
[kaimanovich1983examples] V. A. Kauimanovich, "Examples of nonabelian discrete groups with nontrivial exit boundary," Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), vol. 123, pp. 167-184, 1983.

Show bibtex

@ARTICLE{kaimanovich1983examples, author = {Kaĭmanovich, V. A.}, title = {Examples of nonabelian discrete groups with nontrivial exit boundary}, note = {Differential geometry, Lie groups and mechanics, V}, journal = {Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI)}, fjournal = {Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta imeni V. A. Steklova Akademii Nauk SSSR (LOMI)}, volume = {123}, year = {1983}, pages = {167--184}, issn = {0373-2703}, mrclass = {60B15 (60J15 60J50)}, mrnumber = {0697250}, mrreviewer = {P. Gerl}, zblnumber = {}, }
[kaimanovich1983random] $Go to document$ V. A. Kauimanovich and A. M. Vershik, "Random walks on discrete groups: boundary and entropy," Ann. Probab., vol. 11, iss. 3, pp. 457-490, 1983.

Show bibtex

@ARTICLE{kaimanovich1983random, author = {Kaĭmanovich, V. A. and Vershik, A. M.}, title = {Random walks on discrete groups: boundary and entropy}, journal = {Ann. Probab.}, fjournal = {The Annals of Probability}, volume = {11}, year = {1983}, number = {3}, pages = {457--490}, issn = {0091-1798}, mrclass = {60B15 (43A07 60J15)}, mrnumber = {0704539}, mrreviewer = {Yves Derriennic}, doi = {10.1214/aop/1176993497}, zblnumber = {0641.60009}, }
[mclain1956remarks] D. H. McLain, "Remarks on the upper central series of a group," Proc. Glasgow Math. Assoc., vol. 3, pp. 38-44, 1956.

Show bibtex

@ARTICLE{mclain1956remarks, author = {McLain, D. H.}, title = {Remarks on the upper central series of a group}, journal = {Proc. Glasgow Math. Assoc.}, volume = {3}, year = {1956}, pages = {38--44}, mrclass = {20.0X}, mrnumber = {0084498}, mrreviewer = {K. A. Hirsch}, zblnumber = {0072.25702}, }
[robinson1972finiteness] D. J. S. Robinson, Finiteness Conditions and Generalized Soluble Groups. Part 2, Springer-Verlag, New York, 1972, vol. 63.

Show bibtex

@BOOK{robinson1972finiteness, author = {Robinson, Derek J. S.}, title = {Finiteness Conditions and Generalized Soluble Groups. {P}art 2}, series = {Ergeb. Math. Grenzgeb.}, volume = {63}, publisher = {Springer-Verlag, New York}, year = {1972}, pages = {xiii+254}, mrclass = {20E15}, mrnumber = {0332990}, mrreviewer = {B. Amberg}, zblnumber = {0243.20033}, }
[rosenblatt1981ergodic] $Go to document$ J. Rosenblatt, "Ergodic and mixing random walks on locally compact groups," Math. Ann., vol. 257, iss. 1, pp. 31-42, 1981.

Show bibtex

@ARTICLE{rosenblatt1981ergodic, author = {Rosenblatt, Joseph}, title = {Ergodic and mixing random walks on locally compact groups}, journal = {Math. Ann.}, fjournal = {Mathematische Annalen}, volume = {257}, year = {1981}, number = {1}, pages = {31--42}, issn = {0025-5831}, mrclass = {43A05 (28C10 60B15 60J15)}, mrnumber = {0630645}, mrreviewer = {Yves Guivarc'h}, doi = {10.1007/BF01450653}, url = {https://doi.org/10.1007/BF01450653}, zblnumber = {0451.60011}, }

Choquet-Deny groups and the infinite conjugacy class property

Abstract

Authors