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$\newcommand{\triv}{\mathit{triv}}$A group $G$ has property $T$ if $\triv$ is isolated in the space of unitary representations with the Fell topology.

In general, we heuristically have $H^1(G,Ad(V))$ (where $Ad$ is the kernel of $V \otimes V^* \to triv$) is the tangent space to the representation $V$ in the space of representations.

Thus, I'd expect property $T$ is about $H^1(G,\triv) = 0$.

Instead, I've seen in the literature that the $T$ property for a given representation $V$ (i.e., the existence of compact $K \subset G, \epsilon$ uniformly blocking invariant vectors) is equivalent to $H^1(G,V) = 0$.

How does this align with the tangent space intuition?

Edit-

I am rightfully told that I am completely wrong, but I'm okay with that. I'm trying to understand why the heuristic fails, and what is thus the true reason we use $H^1(G,V)$? How does it relate with the trivial representation being isolated.

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  • $\begingroup$ The Property $H^1(G,\mathrm{triv})$ is much weaker since it just means $\mathrm{Hom}(G,\mathbf{R})=0$. For $\sigma$-compact locally compact groups Property T means $H^1(G,\pi)=0$ for every representation $\pi$. $\endgroup$
    – YCor
    Commented Mar 11 at 15:43
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    $\begingroup$ Your heuristics is wrong even in the finite-dimensional case. $\endgroup$
    – Moishe Kohan
    Commented Mar 11 at 16:16
  • $\begingroup$ The $(2,3,7)$ triangle group is probably the easiest example of a group with $H^1(G,triv)=0$ but without property T. $\endgroup$
    – HJRW
    Commented Mar 11 at 21:00
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    $\begingroup$ @HJRW If you really want the "easiest", I'd say (among f.g. groups) the infinite dihedral group. Or, among countable groups, infinite locally finite ones. $\endgroup$
    – YCor
    Commented Mar 11 at 22:41
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    $\begingroup$ @HJRW in the context of Kazhdan's Property T, it's complex unitaries reps. So $H^1(G,\mathrm{triv})$ means $\mathrm{Hom}(G,\mathbf{C})=0$. For $G$ discrete group it just means the abelianization is torsion. More generally for $G$ locally compact it means the Hausdorff abelianization is an increasing union of compact open subgroups (e.g. $G=\mathbf{Q}_p$ satisfies this). $\endgroup$
    – YCor
    Commented Mar 12 at 7:42

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