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We construct an explicit family of arithmetic Teichm\"uller curves$\mathcal{C}_{2k}$, $k\in\mathbb{N}$, supporting$\textrm{SL}(2,\mathbb{R})$-invariant probabilities $\mu_{2k}$ such that theassociated $\textrm{SL}(2,\mathbb{R})$-representation on $L^2(\mathcal{C}_{2k},\mu_{2k})$ has complementary series for every $k\geq 3$. Actually, the size ofthe spectral gap along this family goes to zero. In particular, theTeichm\"uller geodesic flow restricted to these explicit arithmeticTeichm\"uller curves $\mathcal{C}_{2k}$ has arbitrarily slow rate ofexponential mixing.

Explicit Teichm\"uller curves with complemetary series