Nielsen equivalence in Random groups

10 December 2013
Richard Weidmann
We show that for every $n\ge 2$ there exists a torsion-free one-ended word-hyperbolic group $G$ of rank $n$ admitting generating $n$-tuples $(a_1,\ldots ,a_n)$ and $(b_1,\ldots ,b_n)$ such that the $(2n-1)$-tuples $$(a_1,\ldots ,a_n, \underbrace{1,\ldots ,1}_{n-1 \text{ times}})\hbox{ and }(b_1,\ldots, b_n, \underbrace{1,\ldots ,1}_{n-1 \text{ times}} )$$ are not Nielsen-equivalent in $G$. The group $G$ is produced via a probabilistic construction (joint work with Ilya Kapovich).