Does the following statement hold:
Statement: For any $\epsilon > 0$, there exist a number field $k$ of degree $d_{\epsilon}$ over $\mathbb{Q}$ and an arithmetic hyperbolic surface $\Gamma$ corresponding to an order in a quaternion algebra over $k$ such that $\Gamma$ has a pair of pants whose cuffs are geodesics of length less than $\epsilon d_{\epsilon}$.
Some observations:
0) This geometric condition is equivalent to the existence of two matrices $A, B$ in $\Gamma$ (thought as matrices of $PSL_2(\mathbb{R})$) such that $A, B$ are simultaneous conjugate to matrices $A', B'$, both with matrix norm bounded by $e^{\epsilon' d}$.
1) Because of Lehmer's conjecture, $d_{\epsilon} \to \infty$ as $\epsilon \to 0$.
2) Also, I think one can find such $\Gamma$'s with short curves. (because one can find Salem Polynomials with arbitrary degree and bounded roots).
3) It is known that for an arithmetic surface of genus $g$ one has $d \leq 3\log(g) + 30$ and a surface of genus $g$ can have a systole of length at most $O(\log(g))$, so if one finds hyperbolic surfaces with $d \sim O(\log(g))$, I think the statement have some chances of being true.
I'm also looking for more understanding of the geometry of such surfaces (arithmetic, and in number fields of large degree) and in the same question for arithmetic 3-manifolds or other symmetric spaces (meaning where 0) holds). Any info that looks kind of relevant would be greatly appreciated.
