Group-theoretic compactification of Bruhat–Tits buildings

Abstract

Let G F denote the rational points of a semisimple group G over a non-archimedean local field F, with Bruhat–Tits building X. This paper contains five main results. We prove a convergence theorem for sequences of parahoric subgroups of G F in the Chabauty topology, which enables us to compactify the vertices of X. We obtain a structure theorem showing that the Bruhat–Tits buildings of the Levi factors all lie in the boundary of the compactification. Then we obtain an identification theorem with the polyhedral compactification (previously defined in analogy with the case of symmetric spaces). We finally prove two parametrization theorems extending the Bruhat–Tits dictionary between maximal compact subgroups and vertices of X: one is about Zariski connected amenable subgroups and the other is about subgroups with distal adjoint action.