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It is well-known that amenable groups which have property T are necessarily compact. I am interested if the situation is the same for inner amenable group with property T? Especially if the group is discrete.

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    $\begingroup$ For a discrete I saw two distinct definitions of inner amenable in the literature: existence of a conjugation invariant mean 1) on $G-\{1\}$ 2) on $G$ + atomless. Could you include your definition of "inner amenable"? $\endgroup$
    – YCor
    Aug 24, 2016 at 14:35
  • $\begingroup$ Anyway, a group $G$ has Property FM if any discrete $G$-set with an invariant mean has a finite orbit (and hence the mean is supported by the union of finite orbits). If $G$ has Property FM you can easily check: (1) $G-\{1\}$ has no conjugation invariant mean iff $G$ is icc (all nontrivial conjugacy classes are infinite), i.e. $FC(G)=\{1\}$, where $FC(G)$ is the union of all finite conjugacy classes; (2) $G$ has no atomless invariant mean iff $FC(G)$ is finite. $\endgroup$
    – YCor
    Aug 24, 2016 at 14:41
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    $\begingroup$ In the previous comment, I forgot to say that Property T implies Property FM. In particular, for both meanings of inner amenable, a Property T group with infinite center is inner amenable. $\endgroup$
    – YCor
    Aug 24, 2016 at 20:51
  • $\begingroup$ thank your explanations! but can you please explain more particularly why property T groups with infinite center are inner amenable? $\endgroup$ Aug 25, 2016 at 12:11
  • $\begingroup$ Can you please give me some references to FM property and its connection with property T? $\endgroup$ Aug 25, 2016 at 12:12

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