BIG EDIT of the previous question "Coverings of the free Burnside groups", never answered.

In the paper http://link.springer.com/article/10.1007/BF00046586 (last section) there is an interesting statement concerning the finiteness of the 2-generator Burnside groups $B(2,n)$ (let's assume $n\geq 4$). More precisely, it is claimed that, once $B(2,n)$ is identified with the quotient of the Fuchsian von Dyck group $D(n,n,n)$ (see, e.g. von dyck groups and solvability for its definition) by its $n^\textrm{th}$ powers subgroup $K_n:=D(n,n,n)^n$, then $B(2,n)$ acts on the Riemann surface $\frac{\mathbb{H}}{K_n}$ and, hence, if the latter is compact, the group is finite.

QUESTION 1: what is known about the surface $\frac{\mathbb{H}}{K_4}$? Should be compact, since $B(2,4)$ has order 1024.

(The "planar case" $\frac{\mathbb{R}^2}{K_3}$ is a nice toy model: one obtains a tiled hexagon whose edges correspond to the 27 elements of $B(2,3)$; see, e.g., my paper.)

The above-cited paper is 30-year old, never quoted, and the author himself wouldn't give me explanations. I've found closely related questions on MO (see, e.g., this and this one), but I cannot convince myself about the effectiveness of the proposed algorithm for checking the finiteness of $B(2,n)$, based on the computation of the fundamental domain of $K_n$. In particular, the paper proposes to construct the standard polygon of $K_n$ and check whether its sides close a compact domain.

QUESTION 1 (equivalent formulation): is the fundamental domain of $K_4$ compact?

The proposed algorithm is the following: we fix an origin $o\in\mathbb{H}$, we start to enumerate the elements of $K_n$, and, for any $g\in K_n$ generated by the enumeration, we construct the geodesic line $\ell_g$ perpendicular to the geodesic segment $[o,g\cdot o]$ through its middle point; if the $\ell_g$'s produced so far form a closed polygon, then the algorithm stops.

QUESTION 2 (main): is this algorithm effective? has been used, e.g., to check the still unknown finiteness of $B(2,5)$?

Any link or reference to current algorithmic methods for the two-generator Burnside problem are quite welcome!