Polyhedral compactifications of Bruhat-Tits buildings of quasi-reductive groups

Given a quasi-reductive group $G$ over a local field $k$, using Berkovich geometry, we exhibit a family of $G(k)$-equivariant compactifications of the Bruhat-Tits building $\mathcal B(G, k)$, constructed and investigated by Solleveld and Louren\c{c}o. The compactification procedure consists in mapping the building in the analytification $G^{\mathrm{an}}$ of $G$, then composing this map with the projections from $G^{\mathrm{an}}$ to its (in general non-compact) pseudo-flag varieties $(G/P)^{\mathrm{an}}$, for $P$ ranging among the pseudo-parabolic subgroups of $G$. This generalises previous constructions of Berkovich, then R\'emy, Thuillier and Werner. To define the embedding, we are led to giving a partial extension to the quasi-reductive context of results due to Rousseau on the functoriality of Bruhat-Tits buildings with respect to field extensions, which are of independent interest. Finally, we conclude by investigating the geometry at infinity of these compactifications. The boundaries are shown to be stratified, each stratum being equivariantly homeomorphic to the Bruhat-Tits building of the maximal quasi-reductive quotient of a pseudo-parabolic subgroup.