On the length of nonsolutions to equations with constants in some linear groups (Q6570065)

Let \(\Gamma\) be a group, \(g \in \Gamma\) and let \(\pi_{g}: \Gamma \ast \mathbb{Z} \rightarrow \Gamma\) be the unique homomorphism such that \(\pi_{g}(\Gamma)=1\) and \(\pi_{g}(1)=g\). An element \(w \in \Gamma \ast \mathbb{Z}\) is a mixed identity (or an identity with constants) for \(\Gamma\), if \(w\) is non-trivial but \(\pi_{g}(w)=e_{\Gamma}\) for all \(g \in \Gamma\). The group \(\Gamma\) is called \(\mathsf{MIF}\) (mixed identity-free) if there are no mixed identities for \(\Gamma\) in \(\Gamma \ast \mathbb{Z}\). Given a finite generating set \(S\) for \(\Gamma\), if \(\Gamma\) is \(\mathsf{MIF}\), then for any \(w \in \Gamma \ast \mathbb{Z}\), there exists \(g \in \Gamma\) of minimal word-length \(|g|_{S}\) satisfying \(w(g)\not =e\). Intuitively, the greater this minimal word-length. The goal of the paper under review is to study how \(|g|_{S}\) can depend on the length of \(w\).\N\NThe authors show that for any finite-rank-free group \(\Gamma\), any word-equation in one variable of length \(n\) with constants in \(\Gamma\) fails to be satisfied by some element of \(\Gamma\) of word-length \(O(\log(n))\). By a result of the first author [J. Group Theory 27, No. 1, 31--59 (2024; Zbl 1530.20090)], this logarithmic bound cannot be improved upon for any finitely generated group \(\Gamma\). Beyond free groups, the method developed by the authors (and the logarithmic bound) applies to a class of groups including \(\mathrm{PSL}_{d}(\mathbb{Z})\) for all \(d \geq 2\), and the fundamental groups of all closed hyperbolic surfaces and \(3\)-manifolds. Finally, using a construction designed by \textit{V. Nekrashevych} [Ann. Math. (2) 187, No. 3, 667--719 (2018; Zbl 1437.20038)], the authors exhibit a finitely generated group \(\Gamma\) and a sequence of word-equations with constants in \(\Gamma\) for which every non-solution in \(\Gamma\) is of word-length strictly greater than logarithmic.