Combinatorial and group-theoretic compactifications of buildings

2009/01/27 by Caprace, Pierre-Emmanuel, Lecureux, Jean · 2 citations
#20E42 #20G25 #22E20 #22F50 #51E24 #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.0901.4188

Abstract

Let X be a building of arbitrary type. A compactification C_r(X) of the set Res(X) of spherical residues of X is introduced. We prove that it coincides with the horofunction compactification of Res(X) endowed with a natural combinatorial distance which we call the root-distance. Points of C_r(X) admit amenable stabilisers in Aut(X) and conversely, any amenable subgroup virtually fixes a point in C_r(X). In addition, it is shown that, provided Aut(X)is transitive enough, this compactification also coincides with the group-theoretic compactification constructed using the Chabauty topology on closed subgroups of Aut(X). This generalises to arbitrary buildings results established by Y. Guivarc'h and B. Rémy in the Bruhat--Tits case.

Cited by

• Boundary actions of CAT(0) spaces: topological freeness and applications to C^*-algebras
• Universal C^∗-Algebras from Graph Products: Structure and Applications

Related

• Group-theoretic compactification of Bruhat-Tits buildings
• Martin compactifications of affine buildings
• Compactifications polygonales d'un immeuble affine
• Topological and algebraic properties of universal groups for right-angled buildings
• Bruhat-Tits Theory from Berkovich's Point of View.<br>I - Realizations and Compactifications of Buildings