The Bolza surface $M$ is the closed hyperbolic surface of genus $2$ that can be obtained by identifying the opposite sides of the regular octagon in $\mathbb{H}^2$.

**What two points on $M$ are farthest away from each other?**

Some thoughts:

It is known that $M = \mathbb{H}^2/G$, where $G \cong \left< a, b, c, d \mid abcda^{-1}b^{-1}c^{-1}d^{-1} \right>$. Let us fix a point $p \in M$ and let $\tilde{p}$ be a lift of $p$ in $\mathbb{H}^2$. Consider the Dirichlet region $D_{\tilde{p}}$ of $G$ centered at the point $\tilde{p}$. Clearly, $D_{\tilde{p}}$ is a convex polygon with finite number of sides. Let $\tilde{v}$ be one of the farthest vertices of $D_{\tilde{p}}$ from $\tilde{p}$. Then the projection $v$ of $\tilde{v}$ on $M$ is furthest away from $p$.

If $D_{\tilde{p}}$ is a regular octagon, then the maximal distance from $p$ to a point on $M$ can be computed explicitly (the radius of the circle circumscribing the octagon). However in general $D_{\tilde{p}}$ can be a convex polygon with number of sides from $8$ to $18$, and the question about the distance between two farthest points on $M$ seems not obvious.

One conjecture is that the maximal distance between two farthest points $p, v$ is attained when $D_{\tilde{p}}$ is a regular octagon.