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Kazhdan Property T for a locally compact group is defined in terms of the unitary dual. Namely, a group has property $T$ if the identity is an isolated point in the space of unitary representations with the Fell topology. See https://en.wikipedia.org/wiki/Kazhdan%27s_property_(T) . Some remarkable groups which have property $T$ are given by finitely generated discrete groups with expanders in their Cayley graph, and thus incapable of coarsely embedding into a HIlbert space.

On the other hand, Property $A$ metric spaces (thus also for finitely generated grous via the Cayley graph) is defined in terms of generalizations of Foelner Sets, see http://www.ams.org/notices/200804/tx080400474p.pdf givings conditions for such a group to coarsely embedd in a Hilbert space.

Is there any known dicotomy for finitely presented groups, as for a discrete, finitely generated group having property $T$ and Property $A$ should be finite. What is a good reference to this fact?

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    $\begingroup$ No, linear groups have Property A (Guentner-Higson-Weinberger), and this includes many infinite Property T groups, such as $SL_3(\mathbf{Z})$. $\endgroup$
    – YCor
    Commented Aug 9, 2016 at 23:39
  • $\begingroup$ I am far from an expert in these matters, but in some sense amenability, the Haagerup property and Property (T) are all "equivariant" conditions, definable in terms of the Fourier or Fourier-Stieltjes algebra of a group. Property A, a.k.a. exactness, is not equivariant $\endgroup$
    – Yemon Choi
    Commented Aug 9, 2016 at 23:58
  • $\begingroup$ As @Ycor has pointed out, the emphasis in your first paragraph does not seem quite correct. It is far harder to find groups that do not coarsely embed into Hilbert space, than to find groups with Property (T). $\endgroup$
    – Yemon Choi
    Commented Aug 10, 2016 at 0:00
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    $\begingroup$ @YemonChoi one strength of amenability is that it's also definable in a non-equivariant way (e.g. Følner sets) and in particular it's a coarse invariant (=quasi-isometry invariant) $\endgroup$
    – YCor
    Commented Aug 10, 2016 at 9:29
  • $\begingroup$ @YCor Good point; thank you for the correction/precision $\endgroup$
    – Yemon Choi
    Commented Aug 10, 2016 at 11:44

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