Fernando Alcalde (joint work with Françoise Dal'Bo, Matilde Martínez, and Alberto Verjovsky)

Dynamics of the horocycle flow for homogeneous and non-homogeneous foliations by hyperbolic surfaces.

Congreso: Workshop Geometry and Dynamics of Foliations (Foliations 2014)

ICMAT, Madrid. September 1-5, 2014

Abstract: The aim of this talk is to present some progress towards the understanding of the dynamics of the horocycle flow on compact foliated manifolds by hyperbolic surfaces. This is motivated by a question formulated by Matilde Martnez and Alberto Verjovsky on the minimality of this flow when the action of the ane group generated by the combined action of the geodesic and horocycle fows is minimal too.

Firstly, we shall extend the classical theorem proved by Gustav A. Hedlund in 1936 on the minimality of the horocycle flow on compact hyperbolic surfaces to homogeneous manifolds for the product of PSL(2,R) and any connected Lie group G.  We shall give an elementary proof that does not use the famous Ratner's Orbit-Closure Theorem. We shall also show that this is always the case for homogeneous Riemannian and Lie foliations. This is a joint work with Françoise Dal'Bo.

Examples and counter-examples will take an important place in our talk. They will serve to illustrate our result, as well as a theorem by Martínez and Verjovsky that characterises the minimality of the affine action. We shall use another classical example to briefly describe some work in progress with Dal'Bo, Martínez and Verjovsky in the non-homogeneous case.

Fernando Alcalde

Insertion-tolerance and local isomorphism property for random graphs

Abstract: Any graphed equivalence relation produces a random rooted graph in the sense of Aldous and Lyons, that is, a random variable with values in the space of isomorphism classes of locally finite rooted graphs. We replace this space with the Gromov-Hausdorff space associated with the Cayley graph G of a finitely generated infinite group. This compact (ultra)metric space is endowed with a continuous graphed equivalence relation defined by root moving. Since these two spaces are related by a natural non-expansive map, any random rooted graph with values in G is an example of random rooted graph as defined by Aldous and Lyons. For the abelian free group on two generators, Ghys constructed an unimodular random rooted subtree of G as the continuous hull of a rigid repetitive subtree of G. Other examples of unimodular random rooted graphs obtained from repetitive subgraphs of Cayley graphs are due to Blanc and Lozano Rojo. We show that the continuous hull of any repetitive subgraph in G is negligible for any group-invariant insertion-tolerant bond percolation process on G in the non-trivial supercritical phase. In collaboration with Álvaro Lozano Rojo and Antón C. Vázquez Martínez.

Congreso: Random walks on groups

Amphithéâtre Hermite, Paris. 27 al 31 de enero de 2014

Trabajo conjunto con Á. Lozano Rojo y A. C. Vázquez Martínez

Autores:: H. Nozawa, J.I. Royo Prieto.

Título: Tenseness of Riemannian flows

Congreso: Knots, Manifolds, and Group Actions.

Lugar:Slubice (Polonia)

Año: septiembre 2013