Topology of leaves for minimal laminations by non-simply connected hyperbolic surfaces

We give the topological obstructions to be a leaf in a minimal lamination by hyperbolic surfaces whose generic leaf is homeomorphic to a Cantor tree. Then, we show that all allowed topological types can be simultaneously embedded in the same lamination. This result, together with results of Alvarez-Brum-Mart\'inez-Potrie and Blanc, complete the panorama of understanding which topological surfaces can be leaves in minimal hyperbolic surface laminations when the topology of the generic leaf is given. In all cases, all possible topologies can be realized simultaneously.Comment: 40 pages. 15 figures. Final version. To appear in Groups, Geometry and Dynamic

Foliated Plateau problems and asymptotic counting of surface subgroups

In [17], Labourie initiated the study of the dynamical properties of the space of $k$-surfaces, that is, suitably complete immersed surfaces of constant extrinsic curvature in $3$-dimensional manifolds, which he presented as a higher-dimensional analogue of the geodesic flow when the ambient manifold is negatively curved. In this paper, following the recent work [5] of Calegari--Marques--Neves, we study the asymptotic counting of surface subgroups in terms of areas of $k$-surfaces. We determine a lower bound, and we prove rigidity when this bound is achieved. Our work differs from that of [5] in two key respects. Firstly, we work with all quasi-Fuchsian subgroups as opposed to merely asymptotically Fuchsian ones. Secondly, as the proof of rigidity in [5] breaks down in the present case, we require a different approach. Following ideas outlined by Labourie in [19], we prove rigidity by solving a general foliated Plateau problem in Cartan--Hadamard manifolds. To this end, we build on Labourie's theory of $k$-surface dynamics, and propose a number of new constructions, conjectures and questions.Comment: 38 Pages, 8 Figures, PDF Onl

Groups with infinitely many ends acting analytically on the circle

This article takes the inspiration from two milestones in the study of non minimal actions of groups on the circle: Duminy's theorem about the number of ends of semi-exceptional leaves and Ghys' freeness result in analytic regularity. Our first result concerns groups of analytic diffeomorphisms with infinitely many ends: if the action is non expanding, then the group is virtually free. The second result is a Duminy's theorem for minimal codimension one foliations: either non expandable leaves have infinitely many ends, or the holonomy pseudogroup preserves a projective structure.Comment: We can now make a precise reference to Deroin's work arXiv:1811.10298. 54 pages, 2 figure