On the actions of the mapping class group
on the arc complex, the curve complex, and some other flag complexes (joint with Athanase Papadopoulos)

In this talk, we discussed joint work with Athanase Papadopoulos. We began by describing the Thurston theory of surface diffeomorphisms interms of the action of the mapping class group on the complex of curves. Weexplained how the various pieces of the Thurston decomposition of a surface diffeomorphism, thick domains and annular or thin domains, fit into a flagcomplex, called the complex of domains. Next, we discussed our computationof the group of automorphisms of this complex of domains. We describednongeometric automorphisms of the complex of domains associated to "biperipheral edges" of this complex. We explained how we are ableto construct a natural quotient complex of the complex of domains bycollapsing biperipheral edges to vertices and, thereby, reduce thecomputation in question to computing the group of automorphisms of thisquotient complex. Finally, we explained how we compute the group ofautomorphisms of this quotient complex, the extended mapping class group ofthe surface in question.