Projective structures for maximal representations

Regular octagon of hyperboloids

Start with a Fuchsian representation of the fundamental group of a genus 2 surface into SL(2,ℝ), identifying the sides of a regular hyperbolic octagon with angles π/4. Embed it reducibly into Sp(4,ℝ) through the diagonal representation. The resulting maximal representation preserves a domain of discontinuity in ℝP³.

In this case the domain is the complement of the black hyperboloid and has two connected components. Removing the interiors of the eight colored hyperboloids gives a fundamental domain. It can be constructed either as the boundary of a Dirichlet-Selberg domain or using quadrics attached to the limit map.

Reducible hexagon of hyperboloids

For any Fuchsian representation of a genus g ≥ 2 surface group into SL(2,ℝ), the same reducible embedding into Sp(4,ℝ) gives a maximal representation preserving a domain of discontinuity in ℝP³. Cutting the surface along simple closed curves decomposes it into hexagons. The domain can likewise be decomposed into hexagons of hyperboloids using quadrics attached to the limit map.

Irreducible hexagon of hyperboloids

Composing a Fuchsian representation with the irreducible representation of SL(2,ℝ) into Sp(4,ℝ) produces a Hitchin representation. Hitchin representations are maximal, so their projective domains of discontinuity can also be tiled by hexagons of hyperboloids. This is one such hexagon.

Here the intersection of two intersecting quadrics is topologically a union of two circles, but need not be a union of two projective lines, unlike in the reducible case.

Ideal triangle of hyperboloids

Given any maximal surface group representation and a maximal lamination of the surface, its domain of discontinuity can be decomposed into ideal triangles of hyperboloids, or ideal triangles of photons; see the work of Mazzoli and Viaggi. All these triangles are equivalent to the one shown here under the action of PSp(4,ℝ) ≅ SO₀(2,3).