Automorphisms of the complex of domains

Abstract

In this paper, we study a flag complex which is naturally associated to the Thurston theory of surface diffeomorphisms for compact connected orientable surfaces with boundary. The various pieces of the Thurston decomposition of a surface diffeomorphism, thick domains and annular or thin domains, fit into this flag complex, which we call the complex of domains. The main result of this paper is a computation of the group of automorphisms of this complex. Unlike the complex of curves, introduced by Harvey, for which, for all but a finite number of exceptional surfaces, by the works of Ivanov, Korkmaz, and Luo, all automorphisms are geometric (i.e. induced by homeomorphisms), the complex of domains has nongeometric automorphisms, provided the surface in question has at least two boundary components. These nongeometric automorphisms of the complex of domains are associated to certain edges of the complex which are naturally associated to biperipheral pairs of pants on the surface in question. We project the complex of domains to a natural subcomplex of the complex of domains by collapsing each biperipheral edge onto the unique vertex of that edge which is represented by a regular neighborhood of a biperipheral curve and, thereby, reduce the computation in question to computing the group of automorphisms of this subcomplex, which we call the truncated complex of domains. Finally, we prove that the group of automorphisms of the truncated complex of domains is the extended mapping class group of the surface in question and, obtain, thereby, a complete description of the group of automorphisms of the complex of domains.