Cubulations of closed hyperbolic manifolds
type: Distribution
access URL: https://theses.hal.science/tel-00690334
format: HTML
issued: 2026-05-21T01:54:43.007909
modified: 2026-05-21T01:54:42.898594

test_moissonnage_selune
type: Agent

Cubulations of closed hyperbolic manifolds
type: Dataset
contact point: Organization
  fn: CCSD
keyword: cat0-cube-complexes
keyword: groupes-hyperboliques
keyword: mathmath-gmmathematics-mathgeneral-mathematics-mathgm
keyword: sous-groupes-de-surface
keyword: 3-varietes-hyperboliques-compactes
keyword: infoeu-reposemanticsdoctoralthesis
keyword: cut-and-cross-join-surfaces
keyword: closed-hyperbolic-3-manifolds
keyword: surface-subgroups
keyword: complexes-cubiques-cat0
keyword: spaces-with-walls
keyword: hyperbolic-groups
keyword: espaces-a-murs
keyword: surfaces-coupeees-croisees
keyword: theses
landing page: https://theses.hal.science/tel-00690334
description: This thesis contributes to the study of geometric actions of word-hyperbolic groups on finite dimensional CAT(0) cube complexes. We are mainly interested in the case of fundamental groups of closed hyperbolic manifolds. The philosophy coming from pioneer work of M. Sageev is that a hyperbolic group with sufficiently many quasi-convex codimension one subgroups acts geometrically on a finite dimensional CAT(0) cube complex. We prove a precise criterion for cubulation in the case of closed hyperbolic manifolds, by constructing spaces with walls quasi-isometric to real hyperbolic space. We next focus on the case of three dimensional closed hyperbolic manifolds which are virtually fibered over the circle. In this setting, we use a construction of incompressibly immersed cut-and-cross-join surfaces due to D. Cooper, D. Long and A. Reid that yields surface subgroups of the fundamental group G of the 3-manifold M. By expanding on work of J. Masters and using the structure of the Cannon-Thurston map, we are able to build many quasi-convex surface subgroups of G whose limits sets may be used to separate any pair of distinct points in the boundary of the universal cover of M. As a consequence, G acts geometrically on a finite dimensional CAT(0) cube complex. D. Wise then asks if it is possible that G acts both geometrically and virtually co-specially (in the sense of F. Haglund and D. Wise) on a CAT(0) cube complex. A positive answer would solve the long-standing conjectures that G is large and M has infinite virtual first Betti number. We then explain why finding a virtually embedded cut-and-cross-join surface in a finite cover of M would be enough to solve this problem. Finally, we give some algebraic and then geometric and cohomological sufficient conditions for a given cut-and-cross-join surface to virtually embed.
identifier: NNT: 2012PA112053
issued: 2026-05-21T01:54:42.942506
language: fr
modified: 2026-05-21T01:54:42.942513