I would appreciate an algorithm to list all elements of a given Von Dyck group $D(p,q,r)$, each once, in a format that would allow me to find compositions and inversions within that list. Alternatively, I can of course work with the triangle group, as long as I'm able to filter on the parity of the word length. The individual elements can be listed as strings of the generators $a$, $b$, $c$ of the triangle group (no inverses needed as all three generators have order 2) or of $x$, $y$ of the Von Dyck group (here inverses can be replaced by powers) or elements of some other discrete group that is easier to work with, via some monomorphism. The problem with the former is that I could not find a reliable way to check if two words describe the same group element or not.

I am aware of matrix representations but I am seeking to do this on a computer using floating-point calculations and I am worried by the accumulated error, which may determine two matrices distinct (with an arbitrarily generous error margin) if they were meant to be the same.

To illustrate, I can achieve my goal with the spherical cases, because I know they are either isomorphic to permutation or alternating groups, namely

• $D(2, 3, 3)$ to $A_4$,
• $D(2, 3, 4)$ to $S_4$,
• $D(2, 3, 5)$ to $A_5$,

plus $D(2, 2, n)$ which is isomorphic to $D_n$ that I can also represent by certain permutations (although simpler ways are available for that case). I can easily list the permutations, compose them, compare them exactly and guarantee bijectivity.

All the other groups have infinite order, so I can't generalize this further. My best attempt so far was to march the Cayley graph breath-first, simply assign an unique identifier to each new element and use geometric arguments in Euclidean and hyperbolic space to mark where I have "already been", to prevent double counting. But this is slow, unsystematic, and does not play nice with compositions.

Is there a generalization of the representation by permutations (I would expect permutations of $\mathbb{N}$ to be needed) applicable in the countably infinite cases?

Update: Not a direct generalization, I think, because e.g. in $D(2, 4, 4)$ there are elements of infinite order (translations), which don't exist in $S_{\mathbb{N}}$. But anything of the same idea.