Thursday, September 28, 2023
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All day |
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4:00pm |
09/28/2023 - 4:00pm This is the third lecture in the seminar. Location:
Prospect 204, B-02
09/28/2023 - 4:00pm Let $G$ be a connected semisimple real algebraic group. Let $\theta$ be a non-empty subset consisting of simple roots of $G$. The class of $\theta$-transverse subgroups of $G$ includes all discrete subgroups of rank one Lie groups, $\theta$-Anosov subgroups and their relative versions. For any Zariski dense $\theta$-transverse subgroup $\Gamma$, we introduce the notion of $\theta$-growth indicators and discuss their properties and roles in the study of conformal measures, extending the work of Quint (2003). We also prove that for any $(\Gamma,\psi)$-conformal measure on the $\theta$-boundary, the conical set of $\Gamma$ has measure either $1$ or $0$, depending on whether the $\psi$-Poincare series diverges or not; this extends recent works of Sambarino and of Canary-Zhang-Zimmer proved for special measures supported on the limit set. Our work is new even for $\theta$-Anosov subgroups and answers a question of Sambarino (2022). Applications include an analogue of the Ahlfors measure conjecture: the limit set of a $\theta$-Anosov subgroup is either the whole boundary or of Lebesgue measure zero. When theta is the set of all simple roots, these were previously obtained by Minju Lee-Oh.
This talk is based on joint work with Dongryul Kim and Yahui Wang. Location:
KT801
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