发表机构
University of Toronto Scarborough（多伦多大学士嘉堡校区）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明余紧双曲格群上随机游走的击中测度关于边界Lebesgue测度奇异，扩展了Kosenko-Tiozzo方法，并解决了Kaimanovich-Le Prince奇异性猜想。
AI 中文摘要
设 $\Gamma<\mathrm{Isom}(\mathbb H^n)$，$n\geq2$，为一个余紧格群，其中包含一个在Sageev意义下的凸余紧余维一子群。我们证明，在$\Gamma$上的每一个有限支撑的容许随机游走的击中测度关于$\partial\mathbb H^n$上的Lebesgue测度是奇异的。证明扩展了Kosenko--Tiozzo的方法，将对循环子群的傅里叶分析替换为冯诺依曼维数论证，该论证也适用于非交换子群。作为推论，我们证明了Kaimanovich和Le Prince的奇异性猜想，适用于具有真余紧立方化的余紧双曲格群、余紧Kleinian群，以及具有余维一全测地子格的余紧双曲格群。特别地，我们的结果涵盖了所有余紧双曲反射群、最简类型的余紧算术格群，以及由混合和近亲繁殖构造产生的余紧非算术格群。
英文摘要
Let $Γ<\mathrm{Isom}(\mathbb H^n)$, $n\geq2$, be a cocompact lattice containing a convex cocompact codimension-one subgroup in the sense of Sageev. We prove that every finitely supported admissible random walk on $Γ$ has hitting measure singular with respect to Lebesgue measure on $\partial\mathbb H^n$. The proof extends the method of Kosenko--Tiozzo, replacing Fourier analysis for a cyclic subgroup by a von Neumann dimension argument that also applies to nonabelian subgroups. As corollaries, we prove the singularity conjecture of Kaimanovich and Le Prince for cocompact hyperbolic lattices admitting a proper cocompact cubulation, cocompact Kleinian groups, and cocompact hyperbolic lattices with totally geodesic sublattices of codimension one. In particular, our result covers all cocompact hyperbolic reflection groups, cocompact arithmetic lattices of simplest type and cocompact nonarithmetic lattices arising from the hybrid and inbreeding constructions.
Comments12 pages, comments are welcome
