Principal Investigator: Bruno Klingler
The project is concerned with fundamental questions arising in the representation theory of Kaehler groups and discrete subgroups of Lie groups. The techniques appeal to various fields, from geometric analysis to arithmetic. There are four aspects to the project: 1) study the rigidity and arithmeticity of cocompact complex hyperbolic lattices, for which Margulis's superrigidity theorem does not apply. More generally the PI is trying to construct a reasonable moduli space for non-archimedean representations of Kaehler groups (analogous to Simpson's non-abelian Hodge theory for archimedean representations). 2) study the positivity of a natural kernel on the Bruhat-Tits building of a p-adic simple Lie groups. This would lead to strong results in the representation theory of such groups. 3) find a short-cut to prove that non-arithmetic lattices don't have property T. 4) use ergodic theory to prove some cases of the Andre-Oort conjecture for Shimura varieties.
The concept of "rigidity" underlies many of nature's secrets: certain natural objects (and their mathematical models, like certain curved spaces appearing in general relativity, or certain groups appearing in quantum mechanics) can not be deformed. A fundamental discovery of mathematical analysis was that "rigidity" is essentially a consequence of "arithmeticity": these rigid objects have a simple description in terms of sequences of rational numbers. These numbers provide the cherished invariants of modern physic. This project aims to develop new methods for investigating crucial occurences of such arithmetic structures in some natural symmetry groups, their related spaces and the dynamical systems where they occur.
