CRM: Centro De Giorgi
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Groups in Algebraic Geometry.

Ergodicity of actions of discrete subgroups of semisimple groups and rigidity results on bounded symmetric domains

speaker: Ngaiming Mok (The University of Hong Kong)

abstract: Given a product $G$ of noncompact simple Lie groups, a closed subgroup $H \subset G$ and an irreducible lattice $\Gamma \subset G$, Moore's Ergodicity Theorem implies that the action of $\Gamma$ on the homogeneous space $GH$ is ergodic if and only if $H$ is noncompact. We apply this ergodic phenomenon to rigidity problems on the (finite-volume) quotient $X := \Gamma \backslash \Omega$ of a bounded symmetric domain $\Omega$ of rank $\ge 2$ by a torsion-free irreducible lattice $\Gamma \subset \text{Aut}(\Omega)$, thereby showing that any nontrivial holomorphic map $f: X \to N$ into a quotient $N$ of {\it any} bounded domain $D$ must lift to a holomorphic embedding $F: \Omega \to D$. The proof makes use of Moore's Ergodicity Theorem in conjunction with bounded holomorphic functions which are extremal with respect to Carathéodory-like complex Finsler metrics. As an application of the proof we show that any nontrivial proper holomorphic mapping $f: X \to Z$ of $X$ onto a complex space $Z$ must be an unramified covering unless the fundamental group $\pi1(Z)$ of $Z$ is finite. Regarding the embedding result, on top of the proof using intrinsic complex Finsler metrics there is an alternative proof which makes use of Moore's Ergodicity Theorem in conjunction with non-tangential boundary values of bounded holomorphic functions on $\Omega$. This alternative proof yields stronger results by producing at the same time a left inverse $R: D \to \Bbb Cn$ of the embedding $F$ (i.e., $R \circ F = \text{id}{\Omega}$) as a bounded holomorphic map.

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