Hitchin components for orbifolds

Abstract

We extend the notion of Hitchin component from surface groups to orbifold groups and prove that this gives new examples of higher Teichmüller spaces. We show that the Hitchin component of an orbifold group is homeomorphic to an open ball and we compute its dimension explicitly. We then give applications to the study of the pressure metric, cyclic Higgs bundles, and the deformation theory of real projective structures on 3-manifolds.

Special elliptic isometries, relative SU(2,1)-character varieties, and bendings

“…This means that, in a certain sense, in order to construct a polyhedron with side-pairing isometries that have a chance of succeeding as a fundamental polyhedron, some relations between those isometries of X that will play the role of side-pairing isometries must be known a priori. 1 More generally, the space of representations of the fundamental group π 1 (M ) in some group G of automorphisms of the model space modulo conjugation, i.e., the G-character variety of M , is closely related to the geometric structures on M inherited from the model space. Hence, it is natural to expect that (relative) character varieties are ubiquitous objects in geometry and that the many questions related to its structure (topology, Hitchin components, nature of the action of the mapping class group, etc.)…”

“…are sources of great interest. They have been investigated by several authors, and an exhaustive list of references would be too long to compile; so, we only cite a few ones [1], [7], [9], [11], [13], [14], [16] which are closer to this paper.…”

“…Here, our model space is the complex hyperbolic plane H 2 C with orientation-preserving isometries or, equivalently, the holomorphic 2-ball with its complex automorphisms; the corresponding group is the projective unitary group PU (2,1). A rough classification of nontrivial orientation-preserving isometries in the complex hyperbolic plane resembles that of constant curvature hyperbolic geometry: they either have a fixed point in H 2 C (elliptic isometries), exactly one fixed point in the ideal boundary of H 2 C (parabolic isometries), or exactly two fixed points in this ideal boundary (loxodromic isometries).…”

“…. R p2 R p1 = 1 between holomorphic involutions in PU (2,1). If we move the points p i−1 , p i along a geodesic that joins them without altering their distance, we obtain new points q i−1 , q i satisfying R qi R qi−1 = R pi R pi−1 .…”

“…Special elliptic isometries can be seen as rotations around (negative) points or rotations around complex geodesics (equivalently, around positive points). Since every orientation-preserving isometry has three lifts to SU(2, 1) that differ by a cube root of unity, a nontrivial special elliptic isometry is determined, at the level of SU (2,1), by a (negative or positive) point p, its centre, and by a unitary complex number α distinct from a cube root of unity, its angle. Throughout the paper, we deal with elements in SU (2,1); so, we write a relation between special elliptic isometries in the form R pn αn .…”