Proper isometric actions of hyperbolic groups on $L^p$-spaces
Proper isometric actions of hyperbolic groups on $L^p$-spaces
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$L^p$-空间上双曲群的正确等距作用
DOI：
10.1112/s0010437x12000693
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发表时间：
2012
影响因子：
1.8
通讯作者：
B. Nica
中科院分区：
文献类型：
--
作者：
B. Nica
本文证明了每个非初等双曲群$\G $在$L^p（\bd \G \times \bd \G）$上存在一个真仿射等距作用，其中$\bd \G $表示$\G $的边界，且$p$足够大.我们的建设涉及到$\bd \G \times \bd \G $上的$\G $不变测度类似于从${\rmCAT}（-1）$设置的Bowen-Margulis测度，以及一个几何的Busemann型上循环。我们还证明了当p足够大时，$\G $在第一个$\ell ^p$-上同调群$H^1_{（p）}（\G）$上有一个真仿射等距作用.
Abstract We show that every non-elementary hyperbolic group $\G $ admits a proper affine isometric action on $L^p(\bd \G \times \bd \G )$, where $\bd \G $ denotes the boundary of $\G $ and $p$ is large enough. Our construction involves a $\G $-invariant measure on $\bd \G \times \bd \G $ analogous to the Bowen–Margulis measure from the ${\rm CAT}(-1)$ setting, as well as a geometric, Busemann-type cocycle. We also deduce that $\G $ admits a proper affine isometric action on the first $\ell ^p$-cohomology group $H^1_{(p)}(\G )$ for large enough $p$.