Deformation Theory of infinite-type hyperbolic manifolds

Project Information

DefHyp

Grant agreement ID: 101107744

DOI

10.3030/101107744

Project closed

EC signature date 26 April 2023

Start date 1 July 2023

End date 31 July 2025

Funded under

• Marie Skłodowska-Curie Actions (MSCA)

Total cost

No data

EU contribution

€ 175 920,00

Coordinated by

THE PROVOST, FELLOWS, FOUNDATION SCHOLARS & THE OTHER MEMBERS OF BOARD, OF THE COLLEGE OF THE HOLY & UNDIVIDED TRINITY OF QUEEN ELIZABETH NEAR DUBLIN
Ireland

Periodic Reporting for period 1 - DefHyp (Deformation Theory of infinite-type hyperbolic manifolds)

Reporting period: 2023-07-01 to 2025-07-31

Summary of the context and overall objectives of the project

The DefHyp project aimed to explore the deformation theory of infinite-type hyperbolic 3-manifolds,
building on the researcher's pioneering work in this emerging area of geometry and topology. The
main research objectives, as outlined in Annex 1, include:
1. Parametrize quasi-conformal deformations of hyperbolic structures on infinite-type 3-manifolds
Q1.A).
2. Investigate quasi-conformal rigidity in infinite-type hyperbolic manifolds (Q2).
3. Explore uniqueness and rigidity questions in the spirit of the Mostow Rigidity and Ending
Lamination Theorems (Q3, Q4).
4. Analyze the renormalised volume and its implications for AdS/CFT correspondence and
quantum gravity (Q5, Q6).
Good progress has been made toward Q1 and Q2, culminating in several key results now
published or submitted by the researcher and he has now an almost completed pre-print. These
works lay foundational tools for understanding deformation spaces of infinite-type manifolds,
addressing previously open questions. Progress on Q3 and Q4 has been more limited, but new
research directions have been opened thanks to the participation at the BIRS: Blooming Beast.
Important progress on Q5 has been done culminating in two papers and third one being written.
The second question on renormalized volume has partial result. The researcher and collaborators
have proven most of the result but are trying to prove a stronger convergence, work up to date has
been written in various manuscripts and opened interesting new directions and re-covered more
explicit bounds on classical results of Thurston by further developing the deformation theory of
hyperbolic 3-manifolds.

Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

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.

Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

The DefHyp project advanced the deformation theory of infinite-type hyperbolic 3-manifolds, providing foundational tools for parametrizing and studying quasi-conformal deformations. Key achievements include:
• Development of results on quasi-conformal deformations and rigidity, leading to published and submitted papers.
• Progress on the renormalised volume problem, with two papers completed and a third in preparation, opening new directions at the interface of geometry and mathematical physics (AdS/CFT correspondence).
• Contributions to 3-manifold topology, including a paper resolving a question by Kirby (1990s).
• A total of 7 research papers submitted/accepted, several openly available on arXiv, and two already published.

These results provide the first systematic framework for extending classical tools from finite-type to infinite-type manifolds. Papers are:
1. Fibered 3-manifolds with unique incompressible surfaces – published in JKTR (2024).
2. Knots in circle bundles are determined by their complements – submitted.
3. Density results for the modular group of infinite-type surfaces – submitted.
4. Filling Riemann surfaces by hyperbolic Schottky manifolds of negative volume – accepted at Journal of Geometry and Physics (JGP).
5. Monomial web basis for the SL(N) skein algebra of the twice punctured sphere – submitted to JKTR.
6. Covers of surfaces – accepted at Algebraic and Geometric Topology.
7. Behaviour of the Schwarzian derivative on long complex projective tubes – submitted.