论文标题
组扩展的泊松边界
Poisson boundary of group extensions
论文作者
论文摘要
鉴于有限生成的组，众所周知的稳定性问题询问泊松 - 弗斯滕贝格边界的非平凡性（这等同于存在非恒定有限谐波函数的存在）取决于对组上简单的随机步行的选择。即使在线性群体类别中，这个问题远非被理解。给定一个可正约的组，例如一个可解决的群体，当时没有已知的表征，即使是一个猜想的表征，它是在接受非平凡边界的简单随机行走时。我们为有限生成的特征性$ p $的线性群体提供了具有非平凡边界的组的表征。我们特别证明，稳定性问题在这类组中具有积极的答案。对于特征性$ 0 $的线性群体，我们证明了边界微不足道的足够条件，这并不取决于选择简单的随机步行。我们猜想我们的足够条件也是必要的。我们的论点基于针对组扩展的新比较标准，新的$δ$限制熵估计和边界非平凡性的标准以及边界琐碎的新“谨慎”标准。
Given a finitely generated group, the well-known Stability Problem asks whether the non-triviality of the Poisson-Furstenberg boundary (which is equivalent to the existence of non-constant bounded harmonic functions) depends on the choice of simple random walk on the group. This question was far from being understood even in the class of linear groups. Given an amenable group, e.g. a solvable group, there is no known characterisation, even a conjectural one, of when it admits a simple random walk with non-trivial boundary. We provide a characterisation of groups with non-trivial boundary for finitely generated linear groups of characteristic $p$. We prove in particular that the Stability Problem has a positive answer in this class of groups. For linear groups of characteristic $0$, we prove a sufficient condition for the triviality of the boundary which does not depend on the choice of a simple random walk. We conjecture that our sufficient condition is also necessary. Our arguments are based on a new comparison criterion for group extensions, on new $Δ$-restriction entropy estimates and a criterion for boundary non-triviality, and on a new "cautiousness" criterion for triviality of the boundary.
