Free group algebras in division rings. (Q2909490)

The authors conjecture that a division ring \(D\) that is finitely generated as a division ring and infinite-dimensional over a central subfield \(k\) should contain the group algebra over \(k\) of a free subgroup of \(D^*\) of rank 2 (and hence one of any positive rank). This is a strengthening of separate but related conjectures of Makar-Limanov and Lichtman. They then settle the case where \(k\) is algebraically closed of characteristic zero and \(D\) is the division ring of quotients of the skew polynomial ring \(L[t;s]\), where \(L\) is a function field over \(k\) and \(s\) is a \(k\)-automorphism of \(L\) of infinite order.NEWLINENEWLINE The authors suggest that the techniques of this paper, which are substantially geometric, should extend to proving that for \(D\) as above but without the restriction on \(k\), there should be a finite-dimensional extension division ring \(E\) such that \(E\) contains \(kG\) for some free subgroup \(G\) of \(E^*\) of rank two and hint that they may do this is a subsequent paper.NEWLINENEWLINE Finally the authors point out that the case, where \(k\) is an uncountable field, has been considered, using a different approach, by \textit{J. P. Bell} and \textit{D. Rogalski} in a preprint of 2011.