On the Markoff spectrum on the Hecke group of index six

The discrete part of the Markoff spectrum on the Hecke group of index 6 was determined by A.~Schmidt. In this paper, we study its Markoff and Lagrange spectra after the smallest accumulation point $4\mathrm{/}\sqrt{3}$. We show that both the Markoff and Lagrange spectra below $4\mathrm{/}\sqrt{3}+\epsilon$ have positive Hausdorff dimension for any positive $\epsilon$. We also find maximal gaps and an isolated point in the spectra.