Explicit Teichm\"uller curves with complemetary series

We construct an explicit family of arithmetic Teichm"uller curves ${\mathcal{C}}_{2k}$, $k\in \mathbb{N}$, supporting $\text{SL}(2,\mathbb{R})$-invariant probabilities ${\mu }_{2k}$ such that the associated $\text{SL}(2,\mathbb{R})$-representation on $L^2($\mathcal{C}${2k}, \mu{2k})$hascomplementaryseriesforevery$k\geq 3$. Actually, the size of the spectral gap along this family goes to zero. In particular, the Teichm"uller geodesic flow restricted to these explicit arithmetic Teichm"uller curves ${\mathcal{C}}_{2k}$ has arbitrarily slow rate of exponential mixing.