In geometric group theory, one often studies groups via their actions on some nice geometric spaces. For the mapping class group of a surface S with a finite set of marked points, the arc complex is one such object which has played this role: this is the simplicial complex where vertices are simple arcs, and simplices are spanned by pairwise disjoint arcs. A key rigidity result of Irmak--McCarthy is that any isomorphism between arc complexes in induced by a homeomorphism.
Endowing S with a half-translation structure, a type of singular Euclidean metric, one can obtain the associated saddle connection complex: this is the induced subcomplex of the arc complex where the vertices correspond to saddle connections (straight-line segments between singularities). In this talk, I will discuss similarities and differences between the arc complex and the saddle connection complex. Our main result is that any isomorphism between saddle connection complexes is induced by an affine diffeomorphism between the associated half-translation surfaces.
This is joint work with Valentina Disarlo and Anja Randecker.