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We classify Veech groups of tame non-compact flat surfaces. In particular we prove that all countable subgroups of $G{L}_{+}(2,R$) avoiding the set of mappings of norm less than 1 appear as Veech groups of tame non-compact flat surfaces which are Loch Ness monsters. Conversely, a Veech group of any tame flat surface is either countable, or one of three specific types.

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