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arXiv 2408.034622024-08-06math.GTmath.GRmath.MG

凸余紧对角作用的刚性

Rigidity of convex co-compact diagonal actions

Subhadip Dey, Beibei Liu

备注：: 10 pages

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AI总结：

本文研究真 CAT(0) 空间乘积中凸子集的刚性，证明当群在两个因子空间上的凸余紧作用构成凸余紧对角作用时，在适当条件下这两个作用具有相同的标记长度谱。

摘要：

Kleiner-Leeb 和 Quint 证明了高阶秩对称空间中的凸子集相比于秩 1 对称空间具有更强的刚性。受此启发，我们考虑真 CAT(0) 空间乘积 $X_1\ imes X_2$ 中的凸子集，并证明对于 $X_i$ 上的任意两个凸余紧作用 $ρ_i(Γ)$（其中 $i=1, 2$），如果 $Γ$ 通过 $ρ=(ρ_1, ρ_2)$ 在 $X_1\ imes X_2$ 上的对角作用也是凸余紧的，那么在适当条件下，$ρ_1(Γ)$ 和 $ρ_2(Γ)$ 具有相同的标记长度谱。

英文摘要：

Kleiner-Leeb and Quint showed that convex subsets in higher-rank symmetric spaces are very rigid compared to rank 1 symmetric spaces. Motivated by this, we consider convex subsets in products of proper CAT(0) spaces $X_1\times X_2$ and show that for any two convex co-compact actions $ρ_i(Γ)$ on $X_i$, where $i=1, 2$, if the diagonal action of $Γ$ on $X_1\times X_2$ via $ρ=(ρ_1, ρ_2)$ is also convex co-compact, then under a suitable condition, $ρ_1(Γ)$ and $ρ_2(Γ)$ have the same marked length spectrum.

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