Representations of surface groups in complex hyperbolic space

All Regulators of Flat Bundles are Torsion

“…Let first X be a compact Reimann surface of genus g > 1 and let ρ : π 1 (X) → SU (n, 1) be a representation. The classifying space BSU δ (∞, 1) carries a special elementĉ ∈ H 2 (BSU δ (∞, 1)), introduced and studied by Morita [18], Toledo [28], Corlette [9] and the author [21]. The pull-backp * c ∈ H 2 (X, R) is a secondary class of X, and the number (p * c, [X]) is called the degree of ρ, denoted degρ.…”

Surface group representations with maximal Toledo invariant

“…For more background on the study of maximal representations we refer to [4], [5], [11], [12], [13], [26], [27], [28], [32], [49], [40], and [51].…”

A Measurable Cartan Theorem and Applications to Deformation Rigidity in Complex Hyperbolic Geometry

“…In this case i ρ satisfies the inequality |i ρ | ≤ rank(X ), and representations for which i ρ = rank(X ) are coined maximal. The case where p = 1, that is when Γ is a surface group, is the object of an ongoing study (see [12,13,20,16,18,8,6,7,2]), and in this situation maximal representations lead to new interesting Kleinian groups in higher rank. On the other hand, if p ≥ 2, we expect maximal representations to come from totally geodesic, possibly holomorphic, embeddings, as it is indeed the case when…”