Projections from Furstenberg boundaries onto maximal flats and barycenter maps
Michelle Bucher, Alessio Savini
TL;DR
The paper addresses extending barycenter constructions to higher-rank symmetric spaces by building continuous, $G$-equivariant projections from Furstenberg boundary data to canonical maximal flats, and by deriving barycenter maps on generic boundary configurations. The approach centers on $w$-projections arising from the Iwasawa decomposition, defines $N_{\mathrm{opp}}\to A$ maps $\psi_w$ and their Weyl-group averaged aggregates to obtain $MA$- and $TA$-equivariant boundary-to-flat projections, and then extends these ideas to a symmetric, barycenter map on full-measure domains of $q$-tuples $(G/P)^{(q)}$ with $q\ge 3$. Key contributions include a constructive, multi-formulation framework connecting $(G/P)_{\mathrm{opp}}$, $N_{\mathrm{opp}}$, and $A$, explicit calculations in rank-1 and rank-3 complex cases, and a higher-rank generalization of Cartan’s barycenter that recovers the geometric barycenter in the rank-one hyperbolic setting. These projections provide new tools for rigidity and boundary cocycle constructions in higher rank, with explicit realizations in concrete groups such as $\mathrm{Isom}^+(\mathbb{H}^n)$ and $\mathrm{SL}(3,\mathbb{C})$.
Abstract
Let $G$ be a semisimple connected Lie group of non-compact type with finite center. Let $K<G$ be a maximal compact subgroup and $P<G$ be a minimal parabolic subgroup. For any pair $(F,x)$, where $F$ is a maximal flat in $G/K$ and $x \in G/P$ is opposite to the Weyl chambers determined by $F$, we define a projection $Φ(F, x) \in F$ which is continuous and $G$-equivariant. Furthermore, if $q \geq 3$, we exhibit a $G$-equivariant continuous map defined on an open subset of full measure of the space of $q$-tuples of $(G/P)^q$ with image in $G/K$. When $G$ is the orientation preserving isometries of real hyperbolic space and $q = 3$, we recover the geometric barycenter of the corresponding ideal triangle. All our proofs are constructive.