We conjecture that the set of homogeneous probability measures on the maximal Satake compactification of an arithmetic locally symmetric space $$S=\Gamma \backslash G/K$$ S = Γ \ G / K is compact. More precisely, given a sequence of homogeneous probability measures on S , we expect that any weak limit is homogeneous with support contained in precisely one of the boundary components (including S itself). We introduce several tools to study this conjecture and we prove it in a number of cases, including when $$G={\mathbf{SL}}_3({{\mathbb {R}}})$$ G = SL 3 ( R ) and $$\Gamma ={\mathbf{SL}}_3({{\mathbb {Z}}})$$ Γ = SL 3 ( Z ) .

中文翻译：

---

局部对称空间紧化测度的收敛

我们推测算术局部对称空间 $$S=\Gamma\backslash G/K$$ S = Γ \ G / K 的最大 Satake 紧化的齐次概率测度集是紧的。更准确地说，给定 S 上的一系列齐次概率测度，我们期望任何弱极限都是齐次的，并且支持恰好包含在其中一个边界分量（包括 S 本身）中。我们介绍了几种工具来研究这个猜想，并在许多情况下证明了它，包括当 $$G={\mathbf{SL}}_3({{\mathbb {R}}})$$ G = SL 3 ( R ) 和 $$\Gamma ={\mathbf{SL}}_3({{\mathbb {Z}}})$$ Γ = SL 3 ( Z ) 。