# Residual finite dimensionality of surface groups

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.

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Does it help that surface groups are fully residually free? Or that they are residually $\mathrm{PSL}_2(\mathbb{F}_p)$? – Richard Kent Aug 29 '11 at 16:44
One would need to show that every unitary representation of a surface group is weakly contained in a direct sum of unitary representation which factor through free quotients. This seems to be difficult as well. – Andreas Thom Aug 29 '11 at 17:14
I think it's fairly easy to see that reps which factor through free quotients are dense, at least in some components of the representation variety. (Richard knows much more about this than I do.) Does that help? – HJRW Aug 30 '11 at 9:51
@HW: You are probably talking about the variety of finite-dimensional (non-unitary) representations (for a fixed finite dimension). I was talking about infinite-dimensional unitary representations. As far as I can see, your observation does not help. – Andreas Thom Aug 30 '11 at 12:04