This question is related to the answer I gave to [this MO question](https://mathoverflow.net/questions/210999/infinitesimal-deformations-of-a-fibration). What I'm asking is probably well-known to the experts in the field, and I apologize in advance if this turns out to be trivial.


By [Mostow Rigidity Theorem](https://en.wikipedia.org/wiki/Mostow_rigidity_theorem), the geometry of a hyperbolic manifold $X$ of real dimension at least $3$ is determined by its fundamental group, and hence it is unique. As a corollary, it follows that a complex ball quotient $X$ of (complex) dimension at least $2$ (for instance, a [fake projective plane](https://en.wikipedia.org/wiki/Fake_projective_plane)) has no deformations in the large (i.e. is *rigid*), in other words the base $\textrm{Def}(X)$ of the Kuranishi family of $X$ consists of a single point. 



> 
> **Question.** Is it also true that $X$ is *infinitesimally rigid*, i.e. that $H^1(X, \, T\_X)=0$?
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If $H^1(X, \, T\_X) \neq 0$ then by Mostow's theorem it would follow that $X$ has *obstructed* first order deformations. I guess this is not the case, but I was unable to find a precise reference or statement in the wealth of results concerning deformations (or rigidity) of hyperbolic manifolds.



Infinitesimal deformations of fake projective planes (or ball quotients)
Yes, this is true. It follows from a more general theorem of Calabi and Vesentini on the vanishing of $H^i(X, T\_X)$ (in a suitable range) for $X$ a compact quotient of an irreducible Hermitian symmetric space of non-compact type
(which predates Mostow rigidity). 


The precise reference is Theorem 2 of their paper "On compact, locally symmetric Kähler manifolds" Ann. of Math. (2) 71 1960 472–507.


