Geodesic
Choquet-Deny groups and the infinite conjugacy class property
Joshua Frisch1 ; Yair Hartman2 ; Omer Tamuz3 ; Pooya Vahidi Ferdowsi1
1 Division of Physics, Mathematics and Astronomy, California Institute of Technology, Pasadena, CA
2 Department of Mathematics, Ben-Gurion University of the Negev, Be'er Sheva, Israel
3 Division of the Humanities and Social Sciences and Division of Physics, Mathematics and Astronomy, California Institute of Technology, Pasadena, CA
Annals of mathematics, Tome 190 (2019) no. 1, pp. 307-320

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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.

DOI : 10.4007/annals.2019.190.1.5

Joshua Frisch 1 ; Yair Hartman 2 ; Omer Tamuz 3 ; Pooya Vahidi Ferdowsi 1

1 Division of Physics, Mathematics and Astronomy, California Institute of Technology, Pasadena, CA
2 Department of Mathematics, Ben-Gurion University of the Negev, Be'er Sheva, Israel
3 Division of the Humanities and Social Sciences and Division of Physics, Mathematics and Astronomy, California Institute of Technology, Pasadena, CA 
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Joshua Frisch; Yair Hartman; Omer Tamuz; Pooya Vahidi Ferdowsi. Choquet-Deny groups and the infinite conjugacy class property. Annals of mathematics, Tome 190 (2019) no. 1, pp. 307-320. doi: 10.4007/annals.2019.190.1.5

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