\(\Aut(F_n)\) actions on representation spaces (Q6568816)

In the paper under review, the author investigates the actions of the automorphism group \(\Aut(F_n)\) on \(\mathrm{Epi}(F_n, G)\), where \(F_n\) is a free group of rank \(n\geq 3\), \(G\) is a topological (or algebraic) group and \N\[\N\mathrm{Epi}(F_n, G):=\{f \in \mathrm{Hom}(F_n, G) \ : \ \overline{f(F_n)}=G\}.\N\]\NNote that \(\overline{f(F_n)}\) denotes the closure (or the Zariski closure) of \(f(F_n)\). In particular, the author extends a conjecture of Wiegold, which states that if \(n\geq 3\) and \(G\) is a finite simple group, then the action of \(\Aut(F_n)\) on \(\mathrm{Epi}(F_n, G)\) is transitive. The paper discusses analogous situations for other types of groups, such as compact Lie groups, non-compact simple analytic groups and simple algebraic groups, exploring the conditions under which the epimorphisms are redundant. An epimorphism \(f: F_n \rightarrow G\) is called \textit{redundant} if there exists a non-trivial free product decomposition \(F_n =A *B\) such that the restriction of \(f\) to \(A\) is still an epimorphism. A central question (Question 1.1) is whether, under certain natural assumptions on \(G\), every epimorphism is redundant.\N\NFirst, the author analyzes the compact Lie groups (restricting to the Haar measure), proving that for compact connected semisimple Lie groups, the action of \(\Aut(F_n)\) on redundant epimorphisms is both ergodic and minimal, confirming a conjecture by \textit{W. M. Goldman} [Geom. Funct. Anal. 17, No. 3, 793--805 (2007; Zbl 1139.57002)]. On the other hand, the non-compact semisimple case is more difficult even when restricted to the Haar measure Indeed, for groups like \(\mathrm{SL}(2, \mathbb{R})\), not all epimorphisms are redundant. In this context, \textit{T. Gelander} and \textit{Y. N. Minsky} [Groups Geom. Dyn. 7, No. 3, 557--576 (2013; Zbl 1290.37003)] answered a conjecture of \textit{A. Lubotzky} [in: Geometry, rigidity and group actions. Selected papers based on the presentations at the conference in honor of the 60th birthday of Robert J. Zimmer, Chicago, IL, USA, September 2007. Chicago, IL: University of Chicago Press. 609--643 (2011; Zbl 1266.20045), Theorem 3.2]. Furthermore, the author introduces the concept of \textit{Zariski redundant} representations and formulates the following conjecture (Conjecture 4.2): every Zariski dense representation \(f: F_n \rightarrow G\) is Zariski redundant. In Section 6, the author formulates Conjecture 6.2, which asks if a representation \(f: F_n \rightarrow \mathrm{GL}_d(\mathbb{C})\) maps all primitive elements to torsion elements, then the image \(f(F_n)\) is virtually solvable. Finally, Section 7 discusses primitive-unipotent representations, while Sections 8 and 9 explore the analog of the ``baby Wiegold conjecture'' in the context of algebraic groups and the study of representations \(\rho\) of \(F_n\) when \(\rho(\mathcal{P}_n)\) is contained in a conjugacy class (see Conjecture 9.3).