We consider a planar surface Σ of infinite type which has Thompson's group T as asymptotic mapping class group. We construct the asymptotic pants complex C of Σ and 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.

The braided Thompson group B is an asymptotic mapping class group of a sphere punctured along the standard Cantor set, endowed with a rigid structure. Inspired from the case of finite type surfaces we consider a HatcherThurston cell complex whose vertices are asymptotically trivial pants decompositions. We prove that the automorphism group B 1 2 of this complex is also an asymptotic mapping class group in a weaker sense. Moreover B The study of such automorphisms groups in the pro-finite or pro-unipotent categories seems fundamental in Grothendieck's program. For instance, although the pro-finite pants complexes are still rigid the automorphism group of the corresponding pro-finite curve complexes is a version of the Grothendieck-Teichmüller group ([18] and references there).Simpler versions of this general question concern the solenoids, whose study was started in [2], and then infinite type surfaces corresponding to direct limits. The purpose of this article is to make progress in the second case using the formalism of asymptotically rigid homeomorphisms and braided Thompson groups developed in [9], [10]. A previous result in this direction is the rigidity theorem proved in [8] for an infinite type planar surface related to the Thompson group T (see [4]).In this note we will consider an infinite surface obtained from the sphere by deleting the standard Cantor set from the equator. Since its mapping class group is a topological group, the authors of [9] introduced a smaller subgroup B called asymptotically rigid mapping class group, which was proved to be finitely presented. As its name suggests, one restricts to mapping classes of those homeomorphisms which preserve an extra structure on the surface, but only outside of large enough compact sub-surfaces.There were different but closely related versions of such asymptotically rigid mapping class groups considered independently by Brin ([3]) and Dehornoy ([6], [7]). All of them are usually designed by the generic term of