Central extensions of the Ptolemy-Thompson group and quantized Teichmuller theory

The central extension of the Thompson group $T$ that arises in the quantized Teichm\"uller theory is 12 times the Euler class. This extension is obtained by taking a (partial) abelianization of the so-called braided Ptolemy-Thompson group introduced and studied in \cite{FK2}. We describe then the cyclic central extensions of $T$ by means of explicit presentations.Comment: 26

Asymptotically rigid mapping class groups and Thompson's groups

We consider Thompson's groups from the perspective of mapping class groups of surfaces of infinite type. This point of view leads us to the braided Thompson groups, which are extensions of Thompson's groups by infinite (spherical) braid groups. We will outline the main features of these groups and some applications to the quantization of Teichm\"uller spaces. The chapter provides an introduction to the subject with an emphasis on some of the authors results.Comment: survey 77

The dilogarithmic central extension of the Ptolemy-Thompson group via the Kashaev quantization

Quantization of universal Teichm\"uller space provides projective representations of the Ptolemy-Thompson group, which is isomorphic to the Thompson group $T$. This yields certain central extensions of $T$ by $\mathbb{Z}$, called dilogarithmic central extensions. We compute a presentation of the dilogarithmic central extension $\hat{T}^{Kash}$ of $T$ resulting from the Kashaev quantization, and show that it corresponds to $6$ times the Euler class in $H^2(T;\mathbb{Z})$. Meanwhile, the braided Ptolemy-Thompson groups $T^*$, $T^\sharp$ of Funar-Kapoudjian are extensions of $T$ by the infinite braid group $B_\infty$, and by abelianizing the kernel $B_\infty$ one constructs central extensions $T^*_{ab}$, $T^\sharp_{ab}$ of $T$ by $\mathbb{Z}$, which are of topological nature. We show $\hat{T}^{Kash}\cong T^\sharp_{ab}$. Our result is analogous to that of Funar and Sergiescu, who computed a presentation of another dilogarithmic central extension $\hat{T}^{CF}$ of $T$ resulting from the Chekhov-Fock(-Goncharov) quantization and thus showed that it corresponds to $12$ times the Euler class and that $\hat{T}^{CF} \cong T^*_{ab}$. In addition, we suggest a natural relationship between the two quantizations in the level of projective representations.Comment: 43 pages, 15 figures. v2: substantially revised from the first version, and the author affiliation changed. // v3: Groups M and T are shown to be anti-isomorphic (new Prop.2.32), which makes the whole construction more natural. And some minor changes // v4: reflects all changes made for journal publication (to appear in Adv. Math.

Local geometry of random geodesics on negatively curved surfaces

It is shown that the tessellation of a compact, negatively curved surface induced by a typical long geodesic segment, when properly scaled, looks locally like a Poisson line process. This implies that the global statistics of the tessellation -- for instance, the fraction of triangles -- approach those of the limiting Poisson line process.Comment: This version extends the results of the previous version to surfaces with possibly variable negative curvatur

Alternating Heegaard diagrams and Williams solenoid attractors in 3--manifolds

We find all Heegaard diagrams with the property "alternating" or "weakly alternating" on a genus two orientable closed surface. Using these diagrams we give infinitely many genus two 3--manifolds, each admits an automorphism whose non-wondering set consists of two Williams solenoids, one attractor and one repeller. These manifolds contain half of Prism manifolds, Poincar\'e's homology 3--sphere and many other Seifert manifolds, all integer Dehn surgeries on the figure eight knot, also many connected sums. The result shows that many kinds of 3--manifolds admit a kind of "translation" with certain stability.Comment: 26 pages, 44 figure

(2+1) gravity for higher genus in the polygon model

We construct explicitly a (12g-12)-dimensional space P of unconstrained and independent initial data for 't Hooft's polygon model of (2+1) gravity for vacuum spacetimes with compact genus-g spacelike slices, for any g >= 2. Our method relies on interpreting the boost parameters of the gluing data between flat Minkowskian patches as the lengths of certain geodesic curves of an associated smooth Riemann surface of the same genus. The appearance of an initial big-bang or a final big-crunch singularity (but never both) is verified for all configurations. Points in P correspond to spacetimes which admit a one-polygon tessellation, and we conjecture that P is already the complete physical phase space of the polygon model. Our results open the way for numerical investigations of pure (2+1) gravity.Comment: 35 pages, 22 figure

Origami surfaces for kinetic architecture

This thesis departs from the conviction that spaces that can change their formal configuration through movement may endow buildings of bigger versatility. Through kinetic architecture may be possible to generate adaptable buildings able to respond to different functional solicitations in terms of the used spaces. The research proposes the exploration of rigidly folding origami surfaces as the means to materialize reconfigurable spaces through motion. This specific kind of tessellated surfaces are the result of the transformation of a flat element, without any special structural skill, into a self-supporting element through folds in the material, which gives them the aptitude to undertake various configurations depending on the crease pattern design and welldefined rules for folding according to rigid kinematics. The research follows a methodology based on multidisciplinary, practical experiments supported on digital tools for formal exploration and simulation. The developed experiments allow to propose a workflow, from concept to fabrication, of kinetic structures made through rigidly folding regular origami surfaces. The workflow is a step-by-step process that allows to take a logical path which passes through the main involved areas, namely origami geometry and parameterization, materials and digital fabrication and mechanisms and control. The investigation demonstrates that rigidly folding origami surfaces can be used as dynamic structures to materialize reconfigurable spaces at different scales and also that the use of pantographic systems as a mechanism associated to specific parts of the origami surface permits the achievement of synchronized motion and possibility of locking the structure at specific stages of the folding.A presente tese parte da convicção de que os espaços que são capazes de mudar a sua configuração formal através de movimento podem dotar os edifícios de maior versatilidade. Através da arquitectura cinética pode ser possível a geração de edifícios adaptáveis, capazes de responder a diferentes solicitações funcionais, em termos do espaço utilizado. Esta investigação propõe a exploração de superfícies de origami, dobráveis de forma rígida, como meio de materialização de espaços reconfiguráveis através de movimento. Este tipo de superfícies tesseladas são o resultado da transformação de um elemento plano, sem capacidade estrutural que, através de dobras no material, ganha propriedades de auto-suporte. Dependendo do padrão de dobragem e segundo regras de dobragem bem definidas de acordo com uma cinemática rígida, a superfície ganha a capacidade de assumir diferentes configurações. A investigação segue uma metodologia baseada em experiências práticas e multidisciplinares apoiada em ferramentas digitais para a exploração formal e simulação. Através das experiências desenvolvidas é proposto um processo de trabalho, desde a conceptualização à construção, de estruturas cinéticas baseadas em superfícies dobráveis de origami rígido de padrão regular. O processo de trabalho proposto corresponde a um procedimento passo-apasso que permite seguir um percurso lógico que atravessa as principais áreas envolvidas, nomeadamente geometria do origami e parametrização, materiais e fabricação digital e ainda mecanismos e controle. A dissertação demonstra que as superfícies de origami dobradas de forma rígida podem ser utilizadas como estruturas dinâmicas para materializar espaços reconfiguráveis a diferentes escalas. Demonstra ainda que a utilização de sistemas pantográficos como mecanismos associados a partes específicas da superfície permite atingir um movimento sincronizado e a possibilidade de bloquear o movimento em estados específicos da dobragem