Boundary maps and maximal representations on infinite-dimensional Hermitian symmetric spaces

Abstract

We define a Toledo number for actions of surface groups and complex hyperbolic lattices on infinite-dimensional Hermitian symmetric spaces, which allows us to define maximal representations. When the target is not of tube type, we show that there cannot be Zariski-dense maximal representations, and whenever the existence of a boundary map can be guaranteed, the representation preserves a finite-dimensional totally geodesic subspace on which the action is maximal. In the opposite direction, we construct examples of geometrically dense maximal representation in the infinite-dimensional Hermitian symmetric space of tube type and finite rank. Our approach is based on the study of boundary maps, which we are able to construct in low ranks or under some suitable Zariski density assumption, circumventing the lack of local compactness in the infinite-dimensional setting.

Infinite dimensional representations of orthogonal groups of quadratic forms with finite index

“…Our definition coincides with the one in [17] and with geometric Zariski density in [27, §5] In [8,Lemma 4.2], it is proved that for any group G ⩽ Isom(X ) of a CAT(0) space X , the boundary of the convex closure any G-orbit does not depend on the choice of the orbit. Since the normalizer N (G) of G permutes the G-orbits, this yields a subspace ∆G ⊂ ∂X , namely the convex closure of any orbit, which is N (G)-invariant.…”

“…The rank of a symmetric space with non-positive sectional curvature is defined as the supremum of the dimensions of totally geodesic embedded Euclidean spaces. The symmetric space P 2 (H) has infinite rank and for example, one can find three points that are not contained in any finite dimensional totally geodesic subspace [17,Example 2.6].…”

“…So, by Proposition 5.1, there is a G-equivariant totally geodesic embedding of X G in X K (p, κ) and we denote by X G its image as well. By [17,Lemma 3.11], there is a unique minimal finite dimensional non-degenerate subspace E 0 ⩽ H such that for any x ∈ X G , x ⩽ E 0 . By uniqueness, this linear subspace E 0 is G-invariant.…”

Superrigidity of maximal measurable cocycles of complex hyperbolic lattices

“…where we highlight the dependence on x of the embedding). If n > 1, by the formula (2) and using [DLP,Corollary 7.1], it follows that almost every slice φ x almost maps chains to chains. Hence, by [DLP, Proposition 7.2], for almost every x ∈ X there exists a totally geodesic embedding X x (p, np) ⊂ X (p, ∞) such that φ x (∂H n C ) ⊂ ∂X x (p, np).…”