Quaternionic Extensions of Hyperbolic Toral Automorphisms
Abstract
We construct real-analytic diffeomorphisms of S3× S3≅ SU(2)×SU(2) that lift We study real-analytic word maps of S3× S3≅ SU(2)×SU(2) obtained by evaluating Nielsen automorphisms of the free group F2. Fried studied these word maps and computed their topological entropy. For each maximal torus T⊂SU(2), the product T× 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 S3× S3. 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 H3k((S3)n) is Λk A. For n≥3, 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 S3× S3, where the group Θ7 of homotopy seven-spheres appears in the kernel of the action on homology.
Related
- Outer automorphisms of HNN extensions of abelian groups
- Extreme values and logarithm laws for toral translations
- Sharp Bounds for Toral Eigenfunctions and Threefold Additive Energy on Circles
- Convergence Rate of Birkhoff Average for Toral Quasi-Periodic Rotations and Applications
- Mixing Rates for Analytic Observables: Toral Endomorphisms and Anosov Blaschke Deformations
BibTeX
@misc{verjovsky2026quaternionic,
title = {{Quaternionic Extensions of Hyperbolic Toral Automorphisms}},
author = {Alberto Verjovsky},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.08252},
eprint = {2608.08252},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.08252},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Quaternionic Extensions of Hyperbolic Toral Automorphisms
AU - Alberto Verjovsky
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2608.08252
UR - https://arxiv.org/abs/2608.08252
AB - We construct real-analytic diffeomorphisms of S^3× S^3≅ SU(2)×SU(2) that lift We study real-analytic word maps of S^3× S^3≅ SU(2)×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⊂SU(2), the product T× 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× 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≥3, 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× S^3, where the group Θ_7 of homotopy seven-spheres appears in the kernel of the action on homology.
ER -