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We prove that all finitely generated fully residually free groups (limitgroups) have a sequence of finite dimensional unitary representations that`strongly converge' to the regular representation of the group. Thecorresponding statement for finitely generated free groups was proved byHaagerup and Thorbj{\o}rnsen in 2005. In fact, we can take the unitaryrepresentations to arise from representations of the group by permutationmatrices, as was proved for free groups by Bordenave and Collins.  As for Haagerup and Thorbj{\o}rnsen, the existence of such representationsimplies that for any non-abelian limit group, the Ext-invariant of the reduced$C^{*}$-algebra is not a group (has non-invertible elements)  An important special case of our main theorem is in application to thefundamental groups of closed orientable surfaces of genus at least two. In thiscase, our results can be used as an input to the methods previously developedby the authors of the appendix. The output is a variation of our previous proofof Buser's 1984 conjecture that there exist a sequence of closed hyperbolicsurfaces with genera tending to infinity and first eigenvalue of the Laplaciantending to $\frac{1}{4}$. In this variation of the proof, the systoles of thesurfaces are bounded away from zero and the surfaces can be taken to bearithmetic.

Strongly convergent unitary representations of limit groups