# Arithmetic Fuchsian lattices that are not finite index subgroups of Eichler orders?

Lindenstrauss' proof of AQUE (arithmetic quantum unique ergodicity) assumes that the Fuchsian lattice is an Eichler order or, if I understand it correctly, a finite index subgroup of an Eichler order.

What would be an example of an arithmetic Fuchsian lattice that is not of finite index in an Eichler order?

• Any arithmetic fuchsian group is conjugate to a finite index subgroup of the normalizer of (the image in $\mathrm{PSL}_2(\mathbf R)$ of an Eichler order in some quaternion algebra (see C. Maclachlan and A. W. Reid, The Arithmetic of Hyperbolic 3-Manifolds). These normalizers are often larger (than their Eichler orders), so these are examples. – few_reps Oct 21 '13 at 2:01

If $\Gamma$ is any Fuchsian group derived from a quaternion algebra then there are infinitely many maximal Fuchsian lattices commensurable to $\Gamma$. In particular, since there are up to conjugation only finitely many maximal orders this implies that there are infinitely many examples of arithmetic groups not derived from a quaternion algebra in any given commensurability class.
For an explicit example of such you can take $\Gamma={\rm SL}_2({\mathbb Z})$, a prime $p$ and $$\Gamma_0(p) = \left\{ \begin{pmatrix}a&b\\c&d\end{pmatrix}\in\Gamma:\: p|b\right\}.$$ Then the normalizer $\Gamma'$ of $\Gamma_0(p)$ in ${\rm SL}_2({\mathbb R})$ is not contained in ${\rm SL}_2({\mathbb Q})$ since $$\begin{pmatrix} 0&-\sqrt p\\ (\sqrt p)^{-1}&0\end{pmatrix}$$ belongs to $\Gamma'$.