An action of the cactus group

TL;DR

This paper constructs an action of the big cactus group on Fock-Goncharov's SL_m decorated Teichmuller space, linking algebraic topology with geometric structures of moduli spaces.

Contribution

It introduces a novel action of the big cactus group on a higher rank decorated Teichmuller space, expanding the understanding of symmetries in moduli space geometry.

Findings

Established a new group action on decorated Teichmuller spaces.

Connected the fundamental group of moduli spaces with algebraic structures.

Provided tools for studying symmetries in higher Teichmuller theory.

Abstract

We construct an action of the big cactus group (the fundamental group of the Deligne-Mumford compactification of the moduli space of real curves of genus zero with n undistinguished marked points) on Fock-Goncharov's SL_m analog of the decorated Teichmuller space of ideal n-gons.