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Rich representations and superrigidity

Part of: Complex spaces with a group of automorphisms Holomorphic mappings and correspondences Model theory Automorphic functions Arithmetic problems. Diophantine geometry Lie groups

Published online by Cambridge University Press:  27 January 2025

GREGORIO BALDI ,
NICHOLAS MILLER ,
MATTHEW STOVER Open the ORCID record for MATTHEW STOVER [Opens in a new window]  and
EMMANUEL ULLMO
GREGORIO BALDI
Affiliation:
CNRS, IMJ-PRG, Sorbonne Université, Paris 75005, France (e-mail: baldi@imj-prg.fr)
NICHOLAS MILLER
Affiliation:
Department of Mathematics, University of Oklahoma, Norman, OK 73019, USA (e-mail: nickmbmiller@ou.edu)
MATTHEW STOVER*
Affiliation:
Department of Mathematics, Temple University, Philadelphia, PA 19147, USA
EMMANUEL ULLMO
Affiliation:
I.H.E.S., Bures-sur-Yvette 91440, France (e-mail: ullmo@ihes.fr)
*
e-mail: mstover@temple.edu
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Abstract

We investigate and compare applications of the Zilber–Pink conjecture and dynamical methods to rigidity problems for arithmetic real and complex hyperbolic lattices. Along the way, we obtain new general results about reconstructing a variation of Hodge structure from its typical Hodge locus that may be of independent interest. Applications to Siu’s immersion problem are also discussed, the most general of which only requires the hypothesis that infinitely many closed geodesics map to proper totally geodesic subvarieties under the immersion.

Keywords

complex hyperbolic latticessuperrigidityHodge theory and Mumford–Tate domainsfunctional transcendenceunlikely intersections

MSC classification

Primary: 14G35: Modular and Shimura varieties 22E40: Discrete subgroups of Lie groups 03C64: Model theory of ordered structures; o-minimality 14P10: Semialgebraic sets and related spaces 32H02: Holomorphic mappings, (holomorphic) embeddings and related questions 32M15: Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras

Information

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 45 , Issue 8 , August 2025 , pp. 2249 - 2272
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Copyright
© The Author(s), 2025. Published by Cambridge University Press

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