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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?

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    $\begingroup$ 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. $\endgroup$ – few_reps Oct 21 '13 at 2:01
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I hate to see an easy question without a proper answer, so I'll elaborate a bit on few_reps comment which really contains all there is to say about this question; his reference to Reid and MacLachlan's book also stands for this answer.

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 $\Gamma'$.

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