Nielsen equivalence in a class of random groups (Q2813666)

Let \(G\) be a group. Two \(k\)-tuples \((g_1,\ldots,g_k)\) and \((g_1',\ldots,g_k')\) of group elements are Nielsen equivalent if there is some \(\alpha \in \mathrm{Aut}(F_k)\) and \(\phi\in\mathrm{Hom}(F_k, G)\) such that \(\phi(x_i) = g_i\) and \(\phi\circ\alpha(x_i) = g_i'\) for a fixed basis \((x_1,\ldots,x_k)\) of \(F_k\). \textit{J. Nielsen} showed that \(\mathrm{Aut}(F_k)\) is finitely generated by Nielsen transformations [Math. Ann. 78, 385--397 (1917; JFM 46.0175.01); Math. Ann. 79, 269--272 (1918; JFM 46.0175.02)], characterizing the notion of Nielsen equivalence in terms of the existence of a finite sequence of moves.NEWLINENEWLINEIn general, characterizing Nielsen equivalence tuples is an undecidable problem, even when \(G\) satisfies strong hypotheses on other decision problems. For generating sets \(g_1,\ldots,g_k\) and \(h_1,\ldots,h_k\) of a group \(G\), there is the obvious observation that the \(2k\)-tuples \((g_1,\ldots,g_k,1,\ldots,1)\) and \((1,\ldots,1,h_1,\ldots,h_k)\) are Nielsen equivalent. For tuples of length less than \(2k\), \textit{M. J. Evans} [Contemp. Math. 421, 101--112 (2006; Zbl 1138.20036); in: Ischia group theory 2006. Proceedings of a conference in honour of Akbar Rhemtulla, Ischia, Naples, March 29--April 1, 2006. Hackensack, NJ: World Scientific. 103--119 (2007; Zbl 1170.20021)] gives, for each \(r\), a metabelian group of rank \(k\) which has two generating sets that are not equivalent as \(k+r\) tuples (with identity filling the remainder of the tuple), though \(r\) is significantly less than \(2k\).NEWLINENEWLINEThe main theorem of the article under review is that for each \(k\geq 2\) there exists a torsion-free one-ended word-hyperbolic group \(G\) with rank \(k\) and two minimal generating sets that are not equivalent as \(2k-1\) tuples. The groups are constructed using probabalistic techniques. Briefly, a proposed Nielsen equivalence is examined via Stalings folds. The size of the tuple constrains the rank of the graphs involved, which in turn places combinatorial constrains on the word expression of one generating set in the other. Pairs of generating sets satisfying these constraints are non-generic in an appropriate sense. Starting with a generic pair of words in the free group, the authors give a presentation for a group \(G\) with the desired properties; using the Arzhantseva-Ol'shanskii method [\textit{G. N. Arzhantseva}, Fundam. Prikl. Mat. 3, No. 3, 675--683 (1997; Zbl 0929.20025); \textit{G. N. Arzhantseva} and \textit{A. Yu. Ol'shanskij}, Math. Notes 59, No. 4, 350--355 (1996; Zbl 0877.20021); translation from Mat. Zametki 59, No. 4, 489--496 (1996)] and small cancellation to control the rank and geometry.