Resumen de Geometric and combinatorial aspects of thompson's group t
This PhD thesis is concerned with Thompson's group T. This infinite, finitely presented, simple group is usually seen as a subgroup of the group of dyadic, piecewise linear, orientation-preserving homeomorphisms of the unit circle (piecewise linear T). However, T can also be identified to: 1.- a group of equivalence classes of balanced pairs of finite binary trees (combinatorial T), 2.- a subgroup of piecewise PSL(2,Z), orientation-preserving homeomorphisms of the projective real line (piecewise projective T), and 3.- the asymptotic mapping class group of a fattened complete trivalent tree in the hyperbolic plane (modular T).
The first result shows that the canonical copy of PSL(2,Z) obtained from the piecewise projective T is a non-distorted subgroup of T. For this, one carries over this subgroup to obtain a characterization into combinatorial T, from which the word length of its elements can be estimated. Then, nondistortion follows from the metric properties of T established by Burillo-Cleary-Stein-Taback. As a corollary, T has non-distorted subgroups isomorphic to the free non-abelian group of rank 2.
Furthermore, PSL(2,Z) is also explicitly given in the piecewise linear form.
The second result uses modular T to state that there are exactly f(n) conjugacy classes of elements of order n, where f is the Euler function. Given a torsion element t of T of order n, a dyadic triangulation of the Poincaré disc which is invariant under the action of t modulo a convex polygon with n sides is found.
The third result constructs a minimal simply-connected contractible cellular complex C on which the group T acts by automorphisms. The automorphism group of C is essentially T itself (strictly speaking it is an extension of T by the group of order 2). The cellular complex C can be seen as a generalization of Stasheff's associahedra for an infinitely sided convex polygon. The action of T on C is transitive on vertices and edges and, plus generally, on associahedral type cells in all dimensions.
The final part deals with the first steps of a research project. One uses the geometric interpretation of the 1-skeleton of C in term of dyadic triangulations of the Poincaré disc to define a geometric boundary at infinity. Although the 1-skeleton of C is proved not to be hyperbolic, the construction imitates Gromov's construction of the boundary of hyperbolic spaces, and allows the description of the nature of some of the boundary points.
