Quaternionic Extensions of Hyperbolic Toral Automorphisms

Abstract

We construct real-analytic diffeomorphisms of \(S^3\times S^3\cong \SU(2)\times\SU(2)\) that lift We study real-analytic word maps of \(S^3\times S^3\cong \SU(2)\times\SU(2)\) obtained by evaluating Nielsen automorphisms of the free group \(F_2\). Fried studied these word maps and computed their topological entropy. For each maximal torus \(T\subset\SU(2)\), the product \(T\times T\) is invariant and carries the corresponding toral dynamics. Their union is the commuting locus. We describe its regular part as a twisted conjugacy-sphere bundle over the punctured pillowcase and determine the local topology at the four central singularities. Periodic toral points give periodic conjugacy two-spheres, which obstruct Anosov hyperbolicity on \(S^3\times S^3\). Normalized images of the coordinate three-cycles converge to stable and unstable Ruelle--Sullivan eigencurrents supported on the commuting locus. The construction extends to \(\Sp(1)^n\). The higher-rank commuting locus has dimension \(n+2\), periodic conjugacy two-spheres persist, and the action on \(H_{3k}((S^3)^n)\) is \(Λ^k A\). For \(n\geq3\), IA-automorphisms can act nontrivially on the \(\SU(2)\)-character variety although they are trivial on abelianization. The degree-three eigencurrents again have a Ruelle--Sullivan description; higher exterior-power classes lead to a compactness problem for normal currents. We also relate the nonuniqueness of Nielsen lifts to Fried's computation of the mapping class group of \(S^3\times S^3\), where the group \(Θ_7\) of homotopy seven-spheres appears in the kernel of the action on homology.

Now the paper is 34 pages long. Important suggestions by Andrey Gogolev and Federico Rodríguez Hertz were included