arXiv subjects
Publications and source records attributed to Ariadna Fossas.
We associate to triangulations of infinite type surface a type of flip graph where simultaneous flips are allowed. Our main focus is on understanding exactly when two triangulations can be related by a sequence of flips. A consequence of our results is that flip graphs for infinite type surfaces have uncountably many connected components.
We study arc graphs and curve graphs for surfaces of infinite topological type. First, we define an arc graph relative to a finite number of (isolated) punctures and prove that it is a connected, uniformly hyperbolic graph of infinite diameter; this extends a recent result of J. Bavard to a large class of punctured surfaces. Second, we study the subgraph of the curve graph spanned by those elements which intersect a fixed separating curve on the surface. We show that this graph has infinite diameter and geometric rank 3, and thus is not hyperbolic.
We define and study analogs of curve graphs for infinite type surfaces. Our definitions use the geometry of a fixed surface and vertices of our graphs are infinite multicurves which are bounded in both a geometric and a topological sense. We show that the graphs we construct are generally connected, infinite diameter and infinite rank.
In this paper we construct a cellular complex which is an infinite analogue to Stasheff's associahedra. We prove that it is contractible and state that its (combinatorial) automorphism group is isomorphic to a semi-direct product of R.J. Thompson's group $T$ with $\mathbb{Z}/2\mathbb{Z}$.
We consider a planar surface Σof infinite type which has the Thompson group T as asymptotic mapping class group. We construct the asymptotic pants complex C of Σand prove that the group T acts transitively by automorphisms on it. Finally, we establish that the automorphism group of the complex C is an extension of the Thompson group T by Z/2Z.
In this paper we characterize the elements of PSL(2,Z), as a subgroup of Thompson group T, in the language of reduced tree pair diagrams and in terms of piecewise linear maps as well. Actually, we construct the reduced tree pair diagram for every element of PSL(2,Z) in normal form. This allows us to estimate the length of the elements of PSL(2,Z) through the number of carets of their reduced tree pair diagrams and, as a consequence, to prove that PSL(2,Z) is a non distorted subgroup of T. In particular, we find non-distorted free non abelian subgroups of T.