Algebraic proof of finiteness of von Dyck groups? The von Dyck groups are quotients of the free group on two generators defined as $$D(l, m, n) = \langle  x, y | x^l = y^m = (xy)^n = 1\rangle$$ where $l, m, n \geq 2$. These groups are the symmetries of a particular triangulation of a (possibly non-Euclidean) surface with triangles of angles $\pi/l, \pi/m, \pi/n$, and  are finite if and only if $$\frac{1}{l} + \frac{1}{m} + \frac{1}{n} > 1,$$ in which case the surface is a sphere, which obviously has a finite set of symmetries. (The possible values of $(l, m, n)$ are $(2, 2, n)$ where $n \geq 2$ and $(2, 3, n$) where $3 \leq n \leq 5$, plus permutations.)
There was a question yesterday from someone who asked if $D(2, 3, 5)$ was finite and couldn't find a proof working from the group presentation alone. Is there an argument distinguishing the finite from the infinite von Dyck groups that works purely algebraically, rather than appealing to geometric properties of the group action? I suspect that these arguments would be difficult to find because the finite and infinite groups have superficially similar presentations, but I can't shake the feeling that there should be one.
 A: Consider the Coxeter group 
$$G=G(\ell,m,n)=\langle u,v,w\mid u^2=v^2=w^2=(uv)^\ell=(uw)^m=(vw)^n=1\rangle.$$
Consider the homomorphism onto $\mathbf{Z}/2\mathbf{Z}$ mapping each generator to $1(\neq 0)$. Just using that $u,v,w$ square to $1$, its kernel $K$ is generated by $uv$, $uw$, $vw$. 
Now let's change the presentation changing $(v,w)$ with $(x,y)$, for $x=vu$, $y=uw$ (so $K$ is generated by $(x,y,xy)$. It gives
$$G=\langle u,x,y\mid u^2=(xu)^2=(uy)^2=x^\ell=y^m=(xy)^n=1\rangle,$$
rewritten
$$G=\langle u,x,y\mid u^2=1,\quad x^\ell=y^m=(xy)^n=1,\quad uxu^{-1}=x^{-1},uyu^{-1}=y^{-1}\rangle.$$
We identify the presentation of a semidirect product $\mathbf{Z}/2\mathbf{Z}\ltimes D(\ell,m,n)$. In other words, the index 2 subgroup $K$ of $G$ is isomorphic (through $(x,y)\mapsto (x,y)$) to $D(\ell,m,n)$.
This is not a proof of finiteness, but reduces the finiteness of $D(\ell,m,n)$ to that of the above Coxeter group $G(\ell,m,n)$, and this is (now!(*)) a much more documented result.
(*) I guess von Dyck studied these groups in the 1880s, much before Coxeter groups were introduced.
