$L^2$-cohomology and deformations of the left regular representation

We study deformations of unitary representations $\pi \text{\hspace{0.17em}⁣}:G\to U(H)$ whose coefficients lie in the Hilbert-Schmidt ideal $HS(H)\subset B(H)$. Interesting applications arise for the left-regular representation of surface groups and, more generally, cocompact lattices in the automorphism group of a Fuchsian building of conformal dimension $<2$.