How to enumerate a discrete group of matrices by their Frobenius norm?

Suppose I have a discrete group G<SL2(C),
and it is finitely generated by some known generators.
That is, G=g1,,gn.

The Frobenius norm of a matrix m=(abcd)
is .
The set of elements in G
having the same Frobenius norm is a discrete subset of a compact set,
and therefore is finite.

I'm interested in algorithms for enumerating elements of G
by their Frobenius norm.
That is, find all elements of smallest (nontrivial) Frobenius norm,
then find all elements of next smallest Frobenius norm,
etc.
I have different techniques for different cases using other properties of the groups, but what is an efficient algorithm to do this generally, knowing only that we are given the generators?

Answer

Attribution
Source : Link , Question Author : j0equ1nn , Answer Author : Community

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