Gromov–Thurston manifolds and anti-de Sitter geometry

Monclair, Daniel; Schlenker, Jean-Marc; Tholozan, Nicolas

doi:10.2140/gt.2026.30.2097

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the journal

Ethics and policies

Peer-review process

Submission guidelines

Submission form

Editorial board

Subscriptions

ISSN (electronic): 1364-0380

ISSN (print): 1465-3060

Author index

To appear

Other MSP journals

Gromov–Thurston manifolds and anti-de Sitter geometry

Daniel Monclair, Jean-Marc Schlenker and Nicolas Tholozan

Geometry & Topology 30 (2026) 2097–2156

DOI: 10.2140/gt.2026.30.2097

Abstract

We consider hyperbolic and anti-de Sitter (AdS) structures on $M \times (0, 1)$ , where $M$ is a $d$ -dimensional Gromov–Thurston manifold. If $M$ has cone angles greater than $2 π$ , we show that there exists a “quasifuchsian” (globally hyperbolic maximal) AdS manifold such that the future boundary of the convex core is isometric to $M$ . When $M$ has cone angles less than $2 π$ , there exists a hyperbolic end with boundary a concave pleated surface isometric to $M$ .

Moreover, in both cases, if $M$ is a Gromov–Thurston manifold with $2 k$ pieces (as defined below), the moduli space of quasifuchsian AdS structures (resp. hyperbolic ends) satisfying this condition contains a submanifold of dimension $2 k - 3$ .

When $d = 3$ , the moduli space of quasifuchsian AdS (resp. hyperbolic) manifolds diffeomorphic to $M \times (0, 1)$ contains a submanifold of dimension $2 k - 2$ , and extends up to a “Fuchsian” manifold; that is, an AdS (resp. hyperbolic) warped product of a closed hyperbolic manifold by $ℝ$ .

We then use this construction of quasifuchsian AdS manifolds to obtain new compact quotients of $O (2 d, 2) ∕ U (d, 1)$ . The construction uses an explicit correspondence between quasifuchsian $(2 d + 1)$ -dimensional AdS manifolds and compact quotients of $O (2 d, 2) ∕ U (d, 1)$ , which we interpret as the space of timelike geodesic Killing fields of ${AdS}^{2 d + 1}$ .

Keywords

Gromov-Thurston manifolds, Lorentzian geometry, Anosov representations

Mathematical Subject Classification

Primary: 22E40, 53C50

References

Publication

Received: 3 December 2023

Accepted: 28 September 2024

Published: 31 July 2026

Proposed: Anna Wienhard

Seconded: David Fisher, Urs Lang

Authors

Daniel Monclair
Université Paris-Saclay Laboratoire de Mathématiques d’Orsay Orsay France

Jean-Marc Schlenker
University of Luxembourg, FTSM Department of Mathematics Esch-sur-Alzette Luxembourg

Nicolas Tholozan
CNRS École Normale Supérieure, PSL Paris France

Open Access made possible by participating institutions via Subscribe to Open.

Volume 30, issue 6 (2026)