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In his 1972 paper Sur la cohomologie des groupes topologiques II, Guichardet proved$^\ast$ that (non-abelian) free groups satisfy the following strong converse of property (T): The $1$-cohomology $H^1(\mathbb F_d,\pi)$ is non-zero for every non-zero unitary representation $\pi$ of $\mathbb F_d$.

Is there any established name for groups with this property? And are there any references beyond Guichardet's article that study these groups?

$^\ast$ That's what I gather from secondary sources. I could not get hold of a copy of this article yet.

Previously asked on math.stackexchange, but it was suggested to repost it here.

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    $\begingroup$ This is false as stated, but possibly true if you assume $d\ge 2$ and replace "unitary" with "nonzero unitary". $\endgroup$
    – YCor
    Commented Sep 22, 2023 at 17:02
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    $\begingroup$ (This is false or $d=1$.) This is indeed true if you assume $d\ge 2$. Here's the proof for $d=2$, generators $x,y$, acting as unitaries $T,U$; we can suppose the Hilbert space to be separable. If $1-T$ is not surjective, then choose $v$ not in the image and choose $x$ to act as $\xi\mapsto T\xi+v$. Same if $1-U$ is not surjective. Otherwise $1-T$ and $1-U$ are both surjective, and hence $1-T^*=T^{-1}(T-1)$ is injective, and similarly for $U$, so $T$ and $U$ both fix only $\{0\}$. Hence conjugating $x$ by an arbitrary nontrivial translation yields a fixed-point-free affine action. $\endgroup$
    – YCor
    Commented Sep 22, 2023 at 17:14
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    $\begingroup$ However I wasn't aware of this observation, and I'm not aware that this property has been formulated in general. The only obvious thing I can see is that $F_2\ast G$ satisfies this for every group $G$, and more generally $\mathbf{Z}\ast G$ satisfies this property if and only if $\mathrm{Hom}(G,\mathbf{R})\neq 0$. $\endgroup$
    – YCor
    Commented Sep 22, 2023 at 17:18
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    $\begingroup$ @Ycor Thanks for your comments. I fixed the problems you mentioned in your first comment. I learned about this property from this conference paper by Martin and Valette: zbmath.org/1223.22005. $\endgroup$
    – MaoWao
    Commented Sep 22, 2023 at 17:27
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    $\begingroup$ One more thing: Fundamental groups of surfaces of genus $>1$ have this property at least for finite-dimensional representations. Not sure about the general case. $\endgroup$
    – Moishe Kohan
    Commented Sep 22, 2023 at 19:10

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