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



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> 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 $1$.
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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 $g$.
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> 
> 


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 $S\to X$.




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### 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.



Need a good name for an algorithmic problem in groups that generalizes the conjugacy problem
There is a name for this problem: (the solvability of) a genus 0 quadratic equation over G. Check out Sections 2.1 and 3.3 in <http://arxiv.org/abs/0802.3839>.


This terminology is also consistent with your topological interpretation. Also, whether the set $$
g\_1^G\cdots g\_n^G
$$ contains a commutator is equivalent to the solvability of the genus 1 quadratic equation $$
[x,y]\prod\_{i=1}^n t\_i^{-1}g\_it\_i=1.
$$ in variables $x,y,t\_1,\ldots,t\_n$. This is terrible terminology.


