Primitivity of group rings of non-elementary torsion-free hyperbolic groups (Q1675088)

Let \(G\) be a group, \(R\) a domain with \(|R|\leq |G|\). We say that \(G\) satisfies Property (*) if for each finite subset \(M\) of \(G\setminus\{1\}\), for any \(m\geq2\), there exist distinct \(a,\,b,\,c\in G\) so that if \(\prod_{i=1}^m g_i^{x_i}=1\), \(g_i\in M\), \(x_i\in\{a,\,b,\,c\}\) then \(x_i=x_{i+1}\) for some \(i\). \textit{J. Alexander} and \textit{T. Nishinaka} [J. Algebra 473, 221--246 (2017; Zbl 1406.16024)] proved that if \(G\) has a non-abelian free subgroup \(F\) with \(|F|=|G|\) and \(G\) satisfies Property (*) then \(RG\) is a primitive ring. The author, applying a theorem of Gromov on freeness of the subgroup generated by big powers and its version in [\textit{O. Kharlampovich} and \textit{A. Myasnikov}, J. Eur. Math. Soc. (JEMS) 14, No. 3, 659--680 (2012; Zbl 1273.20042)], shows that if \(G\) is a non-elementary torsion-free hyperbolic group and \(R\) is countable then the hypotheses of the Alexander-Nishinaka theorem [loc. cit.] hold, and hence \(RG\) is primitive. Moreover, the author extends the result to virtually torsion-free groups with trivial FC-centers, which would considerably extend the scope of the statement after an affirmative answer to the long-standing conjecture that every infinite hyperbolic group is virtually torsion-free.