Combinatorial and group-theoretic compactifications of buildings

TL;DR

This paper introduces a new combinatorial compactification for buildings, showing its equivalence to horofunction and group-theoretic compactifications under certain conditions, extending known results to arbitrary buildings.

Contribution

It constructs a novel compactification of the set of spherical residues in buildings and proves its equivalence to existing compactifications, generalizing previous results beyond Bruhat--Tits buildings.

Findings

The compactification coincides with the horofunction compactification with the root-distance.

Points in the compactification have amenable stabilisers in Aut(X).

Under transitivity conditions, it matches the group-theoretic compactification.

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. R\'emy in the Bruhat--Tits case.