论文标题
LP空间上的等距动作：依赖于P的值
Isometric actions on Lp-spaces: dependence on the value of p
论文作者
论文摘要
Answering a question by Chatterji--Druţu--Haglund, we prove that, for every locally compact group $G$, there exists a critical constant $p_G \in [0,\infty]$ such that $G$ admits a continuous affine isometric action on an $L_p$ space ($0<p<\infty$) with unbounded orbits if and only if $p \geq p_G$.在$ L_P $空间上存在适当的连续仿射等距动作也有类似的结果。使用谐波共同体的共同学表示，我们还表明，当线性部分来自保留措施的行动，或者更通常是对von Neumann代数和$ p> 2 $的措施保护行动时，就不会出现这种无限的轨道。我们还证明，在$ l_p $衡量等价的下，这个关键常数$ p_g $的稳定性，回答了一个费舍尔的问题。我们用它来表明，对于每个连接的半胶结谎言组$ g $，对于每个晶格$γ<g $，我们都有$p_γ= p_g $。
Answering a question by Chatterji--Druţu--Haglund, we prove that, for every locally compact group $G$, there exists a critical constant $p_G \in [0,\infty]$ such that $G$ admits a continuous affine isometric action on an $L_p$ space ($0<p<\infty$) with unbounded orbits if and only if $p \geq p_G$. A similar result holds for the existence of proper continuous affine isometric actions on $L_p$ spaces. Using a representation of cohomology by harmonic cocycles, we also show that such unbounded orbits cannot occure when the linear part comes from a measure preserving action, or more generally a state-preserving action on a von Neumann algebra and $p>2$. We also prove the stability of this critical constant $p_G$ under $L_p$ measure equivalence, answering a question of Fisher. We use this to show that for every connected semisimple Lie group $G$ and for every lattice $Γ< G$, we have $p_Γ=p_G$.
