Does there exist a torsion-free group $(G,.)$ with the following property?


There exists finite, non-empty subsets $S\_1,S\_2\subset G$ such that for all $s \in S\_1.S\_2$ there are at least two solutions to the equation $s=s\_1.s\_2$ where $s\_1\in S\_1$ and $s\_2\in S\_2$.


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I imagine that such a group $(G,.)$ does exist. If you've found one, please could you explain how you constructed it also?


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The motivation for this question is to prove that the group ring (for example over $\mathbb{Z}$) of any torsion-free group has no zero divisors, as conjectured by Kaplansky. If such $S\_1$ and $S\_2$ never exist then Kaplansky's conjecture would be true for lots of rings (any ring with no zero divisors in fact).


Thank you in advance.



finite subsets of torsion-free groups
These are the "unique product groups". Not every torsion-free group satisfies this property. See Promislow, S. David, A simple example of a torsion-free, nonunique product group. 
Bull. London Math. Soc. 20 (1988), no. 4, 302–304. 


