Transfinite mutations in the completed infinity-gon

摘要

We introduce mutation along infinite admissible sequences for infinitelymarked surfaces, that is surfaces with infinitely many marked points on theboundary. We show that mutation along such admissible sequences produces apreorder on the set of triangulations of a fixed infinitely marked surface. Weprovide a complete classification of the strong mutation equivalence classes oftriangulations of the infinity- gon and the completed infinity-gonrespectively, where strong mutation equivalence is the equivalence relationinduced by this preorder. Finally, we introduce the notion of transfinitemutations in the completed infinity-gon and show that all its triangulationsare transfinitely mutation equivalent, that is we can reach any triangulationof the completed infinity-gon from any other triangulation via a transfinitemutation.