标题： Furstenberg测度在高阶秩中无限余体积离散子群的奇异性 显示英文标题

标题： Singularity of Furstenberg measure for infinite covolume discrete subroups in higher rank

摘要： 我们考虑在半单李群$G$的离散 Zariski 稠密子群$\Gamma$上的对称随机游走。 我们证明，如果$\Gamma$有无限协体积，则其在$G$的 Furstenberg 边界上的关联击中测度是奇异的。 这与 Furstenberg 将布朗运动离散化到格子的情况形成对比，并且当$G$具有更高秩时，这是此类结果的首次出现。 显示英文摘要

摘要： We consider symmetric random walks on discrete, Zariski-dense subgroups $\Gamma$ of a semisimple Lie group $G$ with Property (T). We prove that if $\Gamma$ has infinite covolume, then the associated hitting measure on the Furstenberg boundary of $G$ is singular. This is in contrast to Furstenberg's discretization of Brownian motion to lattices, and it is the first result of this type when $G$ has higher rank.