Does the next assertion maintain:

Assertion: For any $epsilon > 0$, there exist a quantity area $ok$ of diploma $d_{epsilon}$ over $mathbb{Q}$ and an arithmetic hyperbolic floor  $Gamma$ similar to an order in a quaternion algebra over $ok$ such that $Gamma$ has a pair of pants  whose cuffs are geodesics of size lower than $epsilon d_{epsilon}$.

Some observations:

0) This geometric situation is equal 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 $A’, B’$, each with matrix norm bounded by $e^{epsilon’ d}$.

1) Due to Lehmer’s conjecture, $d_{epsilon} to infty$ as $epsilon to 0$.

2) Additionally, I feel one can discover such $Gamma$‘s with quick curves. (as a result of one can discover Salem Polynomials with  arbitrary diploma and bounded roots).

3) It’s identified that for an arithmetic floor of genus $g$ one has $d leq 3log(g) + 30$ and a floor of genus $g$ can have a systole of size at most  $O(log(g))$, so if one finds hyperbolic surfaces with $d sim O(log(g))$, I feel the assertion have some possibilities of being true. 

I am additionally searching for extra understanding of the geometry of such surfaces (arithmetic, and in quantity fields of enormous diploma) and in the identical query (that means the place 0) holds) for arithmetic 3-manifolds or in different semi-simple Lie teams. Any information that appears type of related can be drastically appreciated.