标题： 关于无限余体积离散子群的奇异性猜想的一篇注记 显示英文标题

标题： A note on the singularity conjecture for infinite covolume discrete subgroups

摘要： 我们考虑在半单李群上的随机游走，其中步长分布的支持作为群生成一个 Zariski 稠密的无限协体积离散子群。 当半单李群具有性质 (T) 时，我们证明在 Furstenberg 边界上的平稳测度与 Lebesgue 测度类是奇异的。 这个结果不需要对步长分布的矩或对称性有任何条件。 当半单李群的秩为一且步长分布具有有限的一阶矩时，我们再次证明在 Furstenberg 边界上的平稳测度与 Lebesgue 测度类是奇异的。 对于一般的半单李群，我们也得到了平稳测度和一般 Patteson-Sullivan 测度奇异性的充分条件。 显示英文摘要

摘要： We consider random walks on semisimple Lie groups where the support of the step distribution generates (as a group) a Zariski dense discrete subgroup of infinite covolume. When the semisimple Lie group has property (T), we show that the stationary measure on the Furstenberg boundary is singular to the Lebesgue measure class. This result does not require any condition on the moment or symmetry of the step distribution. When the semisimple Lie group has rank one and the step distribution has a finite first moment, we again show that the stationary measure on the Furstenberg boundary is singular to the Lebesgue measure class. For general semisimple Lie groups, we also obtain a sufficient condition for the singularity of the stationary measure and a general Patterson-Sullivan measure.