Transfinite mutations in the completed infinity-gon

TL;DR

This paper develops a theory of mutation for infinitely marked surfaces, classifies triangulations of the infinity-gon and completed infinity-gon, and introduces transfinite mutations showing all triangulations are interconnected.

Contribution

It introduces transfinite mutations for the completed infinity-gon and classifies strong mutation equivalence classes of triangulations in these infinite surfaces.

Findings

Mutation along infinite sequences induces a preorder on triangulations.

Complete classification of strong mutation equivalence classes for infinity-gon and completed infinity-gon.

All triangulations of the completed infinity-gon are transfinitely mutation equivalent.

Abstract

We introduce mutation along infinite admissible sequences for infinitely marked surfaces, that is surfaces with infinitely many marked points on the boundary. We show that mutation along such admissible sequences produces a preorder on the set of triangulations of a fixed infinitely marked surface. We provide a complete classification of the strong mutation equivalence classes of triangulations of the infinity- gon and the completed infinity-gon respectively, where strong mutation equivalence is the equivalence relation induced by this preorder. Finally, we introduce the notion of transfinite mutations in the completed infinity-gon and show that all its triangulations are transfinitely mutation equivalent, that is we can reach any triangulation of the completed infinity-gon from any other triangulation via a transfinite mutation.