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# Events up GDAR

First meeting of the network: a three days mini-workshop. The list of participants is:

• Thierry Barbot Université d´Avignon France
• Jairo Bochi Pontificia Universidad Católica de Chile
• Joaquín Brum Universidad de la República Uruguay
• Léo Brunswic Université d´Avignon France
• Mario Jorge Dias Carneiro Universidade Federal de Minas Gerais Brazil
• León Carvajales Universidad de la República Uruguay
• Francois Fillastre Université de Cergy-Pontoise France
• Carlos Maquera Universidade de Sao Paulo Brazil
• Andrés Navas USACH Chile
• Viviane Pardini Valerio Universidade Federal de Sao Joao del-Rei Brazil
• Cristóbal Rivas USACH Chile
• Graham Smith Universidade Federal do Rio de Janeiro Brazil
• Richard Urzúa Universidad Católica del Norte Chile
• Cristobal Rivas USACH Chile
• Julio Sánchez UCN Chile

Start End Title Speaker
Wed 05 July 2017
11:00 12:00 Representaciones de Anosov y descomposiciones dominadas Jairo Bochi
14:00 15:00 Constant curvature -1 3d spaces and Teichmüller theory François Fillastre
16:00 17:00 Acciones libres afines de $$\mathbb Z^p$$ sobre toro $$\mathbb T^q$$ Richard Urzua
Thu 06 July 2017
09:45 10:30 Convex cocompact groups and multiblack-holes: further projects Thierry Barbot
11:00 12:00 Singular (2+1) -dimensional GHMC Minkowski spacetimes Léo Brunswic
14:00 15:00 Some questions concerning groups of piecewise affine and piecewise projective diffeomorphisms Andres Navas Flores
15:30 16:30 On the Anosov character of the Pappus-Schwartz representations Viviane Pardini Valerio
Fri 07 July 2017
10:00 11:00 Some insights in Anosov actions Carlos Alberto Maquera Apaza
11:30 12:30 Constructing constant scalar curvature time functions in (3+1) -dimensional GHMC Minkowski spacetime Graham Andrew Craig Smith

#### Representaciones de Anosov y descomposiciones dominadas

##### by Jairo Bochi on Wed 05 July 2017

The concept of dominated splitting comes from ODE and differentiable dynamics. It turns out that Anosov representations are a manifestation of domination. I will discuss these relations. I will also sketch our proof of a result of Kapovich, Leeb, and Porti stating that only Gromov-hyperbolic groups admit Anosov representations. This talk is based on my joint work with Rafael Potrie and Andrés Sambarino.

#### Constant curvature -1 3d spaces and Teichmüller theory

##### by François Fillastre on Wed 05 July 2017

We briefly review the 3d model spaces of curvature -1, which are hyperbolic space, anti-de Sitter space and co-Minkowski (or half-pipe) space. We then give some examples of relations with Teichmüller space of compact surfaces, mainly focusing on co-Minkowski space. This talk is coming from the surveys arxiv.org/1605.04563 and arxiv.org/1611.01065

#### Acciones libres afines de $$\mathbb Z^p$$ sobre toro $$\mathbb T^q$$

##### by Richard Urzua on Wed 05 July 2017

Toda acción de $$\mathbb Z^p$$ sobre $$\mathbb Z^q$$ que actúa por automorfismos de $$\mathbb Z^q$$, con conjunto de puntos fijos diferente de cero, induce una acción unipotente máxima de $$\mathbb Z^p$$ sobre $$\mathbb Z^q$$´, que determina si la acción original es la parte lineal de una acción afín libre de $$\mathbb Z^p$$ sobre el toro $$\mathbb T^q$$.

#### Convex cocompact groups and multiblack-holes: further projects

##### by Thierry Barbot on Thu 06 July 2017

In this survey talk, I will give some insight on examples of discrete subgroups of PO(2,n) that can be seen as holonomies of conformally flat "multiblack holes" and could be the basis of further works continuing the GDAR project.

#### Singular (2+1) -dimensional GHMC Minkowski spacetimes

##### by Léo Brunswic on Thu 06 July 2017

We present two constructions of polyhedral Cauchy-surfaces in flat globally hyperbolic Cauchy-compact spacetimes. The first construction is inspired by a 1987 paper of Penner on so-called decorated Teichmüller space : the convex hull method. We give a new interpretation of this construction in the context of Cauchy-compact flat spacetimes with BTZ-like singularities giving a bijective map from the moduli space of a marked Cauchy-compact spacetimes with BTZ of linear holonomy to the moduli space of marked closed polyhedral surface. The starting point of the second construction is the inverse of this map ; essentially described by Penner, we give a generalization with the prospect of extending this correspondance to spacetimes with massive particles and BTZ singularities.

#### Some questions concerning groups of piecewise affine and piecewise projective diffeomorphisms

##### by Andres Navas Flores on Thu 06 July 2017

In this talk we will review some recent results concerning obstructions for C^1 actions on the circle of certain groups of homeomorphisms and stress that these questions remain open for certain groups of piecewise-projective homeomorphisms.

#### On the Anosov character of the Pappus-Schwartz representations

##### by Viviane Pardini Valerio on Thu 06 July 2017

The talk will be devoted to the Pappus-Schwartz representation and their recent generalization. The reference for this talk is https://arxiv.org/pdf/1610.04049.pdf

#### Some insights in Anosov actions

##### by Carlos Alberto Maquera Apaza on Fri 07 July 2017

In this talk we will discuss the problem of classification of Anosov actions of abelian (or nilpotent) Lie groups. Our emphasis will be on the case where the action is of codimension one.

#### Constructing constant scalar curvature time functions in (3+1) -dimensional GHMC Minkowski spacetime

##### by Graham Andrew Craig Smith on Fri 07 July 2017

We prove that every (3+1)-dimensional flat GHMC Minkowski spacetime carries a unique foliation by spacelike hypersurfaces of constant scalar curvature. In other words, we prove that every such spacetime carries a unique time function with isochrones of constant scalar curvature. Furthermore, this time function is smooth.

Second global meeting of the network: one day mini-workshop at Federal University of Rio de Janeiro. The list of participants is:

• Thierry Barbot Université d´Avignon France
• Léo Brunswic Centre de Recherche Astrophysique de Lyon, France
• Mario Jorge Dias Carneiro Universidade Federal de Minas Gerais Brazil
• Francois Fillastre Université de Cergy-Pontoise France
• Carlos Maquera Universidade de Sao Paulo Brazil
• Andrés Navas USACH Chile
• Viviane Pardini Valerio Universidade Federal de Sao Joao del-Rei Brazil
• Uira Matos Universidade de Sao Paulo Brazil
• Graham Smith Universidade Federal do Rio de Janeiro Brazil
• Sébastien Alvarez Udelar Montevideo Uruguay

Start End Title Speaker
Fri 10 August 2018
09:45 10:30 Convex cocompact groups and multiblack-holes: further projects Thierry Barbot
10:30 11:15 Multicontact Anosov actions Uirá Norberto Matos de Almeida
11:15 11:45 Coffee break -----
11:45 12:30 Random walk on groups (of hyperbolic nature) Victor Guerasimov
13:30 14:15 Poster session -----
14:15 15:00 Inhomogeneous dark energy via Gauss Bonnet Formula Léo Brunswic
15:00 15:45 Other pappus and their representations Heleno da Silva Cunha
15:45 16:15 Coffee break -----
16:15 17:00 TBA Sébastien Alvarez
17:00 18:00 Concluding session -----

#### Convex cocompact groups and multiblack-holes: further projects

##### by Thierry Barbot on Fri 10 August 2018

In this survey talk, I will give some insight on examples of discrete subgroups of PO(2,n) that can be seen as holonomies of conformally flat "multiblack holes" and could be the basis of further works continuing the GDAR project.

TBA

TBA

#### Inhomogeneous dark energy via Gauss Bonnet Formula

##### by Léo Brunswic on Fri 10 August 2018

I will present Buchert's model of inhomogeneous dark energy/matter in all dimension and how it includes the standard cosmological model. Then I will show how this framework together with Gauss-Bonnet formula allows to obtain a dark energy source of topological origin in dimension 2+1. A discussion will follow comparing this result to Mess, Barbot, Bonsante Theorems in higher dimensions.

#### Other pappus and their representations

##### by Heleno da Silva Cunha on Fri 10 August 2018

From the works of Barbot-Lee-Valério and Schwartz we will investigate the theorem of Pappus in other spaces and the representations induced by him. Let us present a new demonstration of Pappus's theorem in the real projective plane and which, with due adaptations, also applies to the complex projective plane and the real and complex hyperbolic planes. Finally, we will discuss the representations in these new "ecosystems".

TBA