Invariant subspaces for some surface groups acting on A2-Euclidean buildings

Abstract

This paper deals with non-Archimedean representations of punctured surface groups in PGL(3), associated actions on Euclidean buildings (of type A2), and degenerations of real convex 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 A2-complex. We use this result to describe degenerations of real convex projective structures on surfaces for an open cone of parameters. The main tool is a geometric interpretation of Fock-Goncharov parametrization in A2-buildings.