Sur la rigidité de certains groupes fondamentaux, l’arithméticité des réseaux hyperboliques complexes, et les “faux plans projectifs”

摘要

The motivation of this work comes from the study of lattices in real simple Lie groups. The famous Margulis’s superrigidity theorem claims that finite dimensional reductive representations of any lattice of a real simple Lie group of real rank ≥2 are superrigid. As a corollary such a lattice is arithmetic. These results extend to the real rank one case for lattices in Sp(n,1) and F 4 (-20) by the work of Corlette and Gromov-Schoen. On the other hand Mostow and Deligne-Mostow exhibited arithmetic lattices with non-superrigid representations as well as non-arithmetic lattices in the unitary group PU(2,1). A natural question is then to find simple sufficient conditions for superrigidity or arithmeticity of lattices in PU(2,1). Rogawski conjectured the following: let Γ be a torsion-free cocompact lattice in PU(2,1) such that the hyperbolic quotient M=ΓB 2 ℂ verifies the cohomogical conditions b 1(M)=0 and H 1,1(M,ℂ)∩H 2(M,ℚ)≃ℚ. Then Γ is arithmetic. In this paper we consider a smooth complex projective surface M verifying the above cohomological assumptions and study Zariski-dense representations of the fundamental group π1(M) in a simple k-group H of k-rank ≤2 (where k denotes a local field). Our main result states that there are strong restrictions on such representations, especially when k is non-archimedean (Theorem 5). We then consider some cocompact lattices in PU(2,1) of special geometric interest: recall that a “fake P 2ℂ” is a smooth complex surface (distinct from P 2ℂ) having the same Betti numbers as P 2ℂ. “Fake P 2ℂ” exist by a result of Mumford and are complex hyperbolic quotients ΓH 2 ℂ by Yau’s proof of the Calabi conjecture. They obviously verify the hypotheses of Rogawski’s conjecture. In this case we prove that every Zariski-dense representation of Γ in PGL(3) is superrigid in the sense of Margulis (Theorem 3). As a corollary every “fake P 2ℂ” is an arithmetic quotient of the ball B 2 ℂ.