Invariant weakly convex cocompact subspaces for surface groups in $A_2$-buildings

Abstract

This paper deals with non-Archimedean representations of punctured surface groups in $\operatorname {PGL}_3$, associated actions on (not necessarily discrete) Euclidean buildings of type $A_2$, and degenerations of convex real projective structures on surfaces. The main result is that, under good conditions on Fock-Goncharov generalized shear parameters, non-Archimedean representations acting on the Euclidean building preserve a cocompact weakly convex subspace, which is part flat surface and part tree. In particular the eigenvalue and length(s) spectra are given by an explicit finite $A_2$-complex. We use this result to describe degenerations of convex real projective structures on surfaces for an open cone of parameters. The main tool is a geometric interpretation of Fock-Goncharov parameters in $A_2$-buildings.