Quasi-Fuchsian AdS representations are Anosov

In a recent paper, Q. M\'erigot proved that representations in SO(2,n) of uniform lattices of SO(1,n) which are Anosov in the sense of Labourie are quasi-Fuchsian, i.e. are faithfull, discrete, and preserve an acausal subset in the boundary of anti-de Sitter space. In the present paper, we prove the reverse implication. It also includes: -- A construction of Dirichlet domains in the context of anti-de Sitter geometry, -- A proof that spatially compact globally hyperbolic anti-de Sitter spacetimes with acausal limit set admit locally CAT(-1) Cauchy hypersurfaces

Broken Lefschetz fibrations and mapping class groups

The purpose of this note is to explain a combinatorial description of closed smooth oriented 4-manifolds in terms of positive Dehn twist factorizations of surface mapping classes, and further explore these connections. This is obtained via monodromy representations of simplified broken Lefschetz fibrations on 4-manifolds, for which we provide an extension of Hurwitz moves that allows us to uniquely determine the isomorphism class of a broken Lefschetz fibration. We furthermore discuss broken Lefschetz fibrations whose monodromies are contained in special subgroups of the mapping class group; namely, the hyperelliptic mapping class group and in the Torelli group, respectively, and present various results on them which extend or contrast with those known to hold for honest Lefschetz fibrations. Lastly, we show that there are 4-manifolds admitting infinitely many pairwise nonisomorphic relatively minimal broken Lefschetz fibrations with isotopic regular fibers.Comment: 18 pages, 3 figure

Orderable 3-manifold groups

We investigate the orderability properties of fundamental groups of 3-dimensional manifolds. Many 3-manifold groups support left-invariant orderings, including all compact P^2-irreducible manifolds with positive first Betti number. For seven of the eight geometries (excluding hyperbolic) we are able to characterize which manifolds' groups support a left-invariant or bi-invariant ordering. We also show that manifolds modelled on these geometries have virtually bi-orderable groups. The question of virtual orderability of 3-manifold groups in general, and even hyperbolic manifolds, remains open, and is closely related to conjectures of Waldhausen and others.Comment: 37 pages. Published version. Improvements in the organisation and presentation of the materia

Spectra of sub-Dirac operators on certain nilmanifolds

We study sub-Dirac operators that are associated with left-invariant bracket-generating sub-Riemannian structures on compact quotients of nilpotent semi-direct products $G=\mathbb{R}^n\rtimes_A\mathbb{R}$. We will prove that these operators admit an $L^2$-basis of eigenfunctions. Explicit examples show that the spectrum of these operators can be non-discrete and that eigenvalues may have infinite multiplicity