The DefHyp project advanced the deformation theory of infinite-type hyperbolic 3-manifolds, an area previously unexplored compared to the finite-type setting. The research was structured around six core questions (Q1–Q6) addressing quasi-conformal deformations, rigidity phenomena, uniqueness theorems, and the study of renormalised volume.
Work carried out included:
• Work Package 1 – Preliminary Work: Completed a thorough literature review in hyperbolic geometry, deformation theory, and renormalised volume, which established the foundation for subsequent results.
• Work Package 2 – Quasi-conformal Deformations: Developed new tools to parametrize quasi-conformal deformations of infinite-type 3-manifolds. This culminated in a preprint and a submitted paper, significantly extending classical approaches.
• Work Package 3 – Rigidity and Uniqueness: Began investigations on Mostow-type and Ending Lamination-type rigidity phenomena for infinite-type manifolds. While still ongoing, exploratory results were achieved and a paper addressing a longstanding question by Kirby (1990s) was completed.
• Work Package 4 – Renormalised Volume: Conducted in-depth analysis of renormalised volume in infinite-type hyperbolic manifolds, producing two completed papers (with one accepted at JGP) and a third long manuscript (~50 pages) in preparation. This work links geometric structures with applications in AdS/CFT correspondence and quantum gravity.
• Collaborations and Mobility: Engaged with leading experts (e.g. Bromberg, Bridgeman, Minsky) through visits, secondments, and workshops, opening new directions in infinite-type Teichmüller theory and big mapping class groups.
Main scientific achievements and outcomes:
• Produced seven research papers, of which two are already published/accepted and the others submitted or available on arXiv.
• Derived new insights on the renormalised volume invariant, offering connections to mathematical physics.
• Opened promising new directions in 3-manifold topology, including the partial resolution of a problem left open since the early 1990s.
In summary, the DefHyp project achieved significant advances in the understanding of infinite-type hyperbolic 3-manifolds, delivering both published results and ongoing lines of inquiry that will shape further research in geometry, topology, and mathematical physics.
