Strongly convergent unitary representations of limit groups

We prove that all finitely generated fully residually free groups (limit groups) have a sequence of finite dimensional unitary representations that ‘strongly converge’ to the regular representation of the group. The corresponding statement for finitely generated free groups was proved by Haagerup and Thorbjørnsen in 2005. In fact, we can take the unitary representations to arise from representations of the group by permutation matrices, as was proved for free groups by Bordenave and Collins.As for Haagerup and Thorbjørnsen, the existence of such representations implies that for any non-abelian limit group, the Ext-invariant of the reduced C⁎-algebra is not a group (has non-invertible elements).