I am looking for a good name for the following problem:

Given elements $g_1,\dotsc,g_n$ in a (finitely generated) group $G$, determine if the product of their conjugacy classes $g_1^G\dotsb g_n^G$ contains the identity element $1$.

In some situations it might be more natural to pose this problem slightly differently:

Given elements $g_1,\dotsc,g_n$ and $g$ in $G$, determine if $g_1^G\dotsb g_n^G$ contains $g$.

This problem generalizes the *conjugacy problem*.

Geometrically the problem can be stated as follows: given a path-connected space $X$ (with fundamental group $G$) and a sphere with holes $S$, determine for every mapping $\partial S\to X$ (a mapping of the boundary circles of $S$ to $X$) if this mapping can be extended to a mapping $S\to X$.

It would be nice to also have consistent names for other related problems. In particular, for this one:

Given elements $g_1,\dotsc,g_n$ in $G$, determine if the product of their conjugacy classes $g_1^G\dotsb g_n^G$ contains a commutator.

Geometrically this problem can be stated as follows: given a path-connected space $X$ and a torus with holes $S$, determine for every mapping $\partial S\to X$ if this mapping can be extended to a mapping $S\to X$.

### Suggestions

Would ** generalized conjugacy problem** be an acceptable name for this problem? I've seen that this term is already used for other problems, but it does not seem to have a generally accepted meaning. For example i've seen it used for the

*simultaneous conjugacy problem*, but we do not need two names for the same problem.

factorization$\endgroup$