On the action of the mapping class group
on the arc complex(joint with Elmas Irmak)

This talk concerns joint work with Elmas Irmakon the action of the mapping class group of a compactconnected orientable surface with boundary on the simplicialcomplex whose simplices are collections of isotopy classesof pairwise disjoint and nonisotopic properly embedded essential arcs onthis surface.

In this talk, we discussed joint work with Elmas Irmak regarding the action of the extended mapping class group of a compact connected orientable surface with nonempty boundary on the arc complex of this surface. After explaining the setting of our study, we gave a number of examples illustrating our main result relating this action to the group of automorphisms of this complex. In particular, we discussed the examples of (i) a pair of pants, which corresponds to the action of the symmetric group on three symbols on the tiling of an equilateral triangle by four equilateral triangles, and (ii) the example of a torus with one hole, which corresponds to the action of PGL(2,Z) on the classical decomposition of the hyperbolic plane by the ideal triangles of the Farey graph. We then outlined the main ideas involved in the proof of our primary result that the natural representation of the extended mapping class group into the group of automorphisms of the arc complex is surjective, with kernel equal to the center of the extended mapping class group.