Alex Lubotzky and Yehuda Shalom have shown in Finite representations in the unitary dual and Ramanujan groups., (Discrete geometric analysis, 173–189, Contemp. Math., 347, Amer. Math. Soc., Providence, RI, 2004.) that every unitary representation of a surface group is weakly contained in a direct sum of unitary representations which factor over finite quotients. This property is called FD.
It clearly implies property RFD (residually finite dimensionality), which says that every unitary representation is weakly contained in a direct sum of finite-dimensional unitary representations.
Question: Is there a direct proof of this consequence?
I am thinking of a direct argument like the classical one for free groups, which is due to Man-Duen Choi, The full $C^∗$-algebra of the free group on two generators. (Pacific J. Math. 87 (1980), no. 1, 41–48.)
Note that FD and RFD are rather subtle properties of the unitary dual of a discrete group. They are not implied by residual finiteness of the group, e.g. for $SL(3,\mathbb Z)$ by a result of Bekka.
