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Quotients of the holomorphic 2-ball and the turnover

Hugo C. Botós, Carlos H. Grossi

TL;DR

The authors develop two-parameter families of complex hyperbolic disc orbibundles over sphere orbifolds with three cone points by realizing turnover groups into $ ext{PU}(2,1)$ and using quadrangles of bisectors to construct discrete actions on $ ext{H}^2_{ ext{C}}}^2$. They show that, beyond previously known locally rigid examples, there exist non-rigid, deformation-rich structures, including complex hyperbolic structures on trivial and cotangent bundles. Central to their approach are the Euler number $e$, the Toledo invariant $ au$, and the orbifold Euler characteristic $oldsymbololdsymbol extchi$, linked by $2(e+oldsymbololdsymbol extchi)=3 au$ in all constructed cases, with $3 au=2(e+oldsymbololdchi)$ arising from holomorphic sections in the presence of a holomorphic structure. The paper combines rigorous construction via turnover representations, explicit parametrizations $(s,t)$, and computational exploration to produce thousands of discrete non-rigid examples, including explicit cotangent-bundle and trivial-bundle cases, expanding the catalog of known complex hyperbolic disc orbibundles and highlighting new geometric-topological phenomena in complex hyperbolic geometry.

Abstract

We construct two-dimensional families of complex hyperbolic structures on disc orbibundles over the sphere with three cone points. This contrasts with the previously known examples of the same type, which are locally rigid. In particular, we obtain examples of complex hyperbolic structures on trivial and cotangent disc bundles over closed Riemann surfaces.

Quotients of the holomorphic 2-ball and the turnover

TL;DR

The authors develop two-parameter families of complex hyperbolic disc orbibundles over sphere orbifolds with three cone points by realizing turnover groups into and using quadrangles of bisectors to construct discrete actions on . They show that, beyond previously known locally rigid examples, there exist non-rigid, deformation-rich structures, including complex hyperbolic structures on trivial and cotangent bundles. Central to their approach are the Euler number , the Toledo invariant , and the orbifold Euler characteristic , linked by in all constructed cases, with arising from holomorphic sections in the presence of a holomorphic structure. The paper combines rigorous construction via turnover representations, explicit parametrizations , and computational exploration to produce thousands of discrete non-rigid examples, including explicit cotangent-bundle and trivial-bundle cases, expanding the catalog of known complex hyperbolic disc orbibundles and highlighting new geometric-topological phenomena in complex hyperbolic geometry.

Abstract

We construct two-dimensional families of complex hyperbolic structures on disc orbibundles over the sphere with three cone points. This contrasts with the previously known examples of the same type, which are locally rigid. In particular, we obtain examples of complex hyperbolic structures on trivial and cotangent disc bundles over closed Riemann surfaces.

Paper Structure

This paper contains 28 sections, 11 theorems, 201 equations, 35 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Complex hyperbolic generalities
  4. Totally geodesic subspaces
  5. Bisectors
  6. Holomorphic isometries
  7. The turnover and its PU(2,1)-character variety
  8. The turnover
  9. Character variety
  10. Generic representations
  11. Proof of Proposition \ref{['prop main result']}
  12. Regular case: $I_2$ is regular elliptic.
  13. Special case: $I_2$ is a rotation about a point in $\mathbb{H}_\mathbb{C}^2$.
  14. Special case: $I_2$ is a rotation about a complex geodesic in $\mathbb{H}_\mathbb{C}^2$.
  15. Discreteness: fundamental quadrangle of bisectors
  16. ...and 13 more sections

Key Result

Proposition 9

If at least two of the $I_j$'s are special elliptic isometries, then $\rho$ stabilizes a projective line.

Figures (35)

  • Figure 1: Bisector foliated by complex geodesics.
  • Figure 2: Meridional decomposition.
  • Figure 5: Quadrangle $\mathcal{Q}$
  • Figure 6: Orienting the real spine fixes a unique orientation of the bisector since the slices are naturally oriented. The half-space $K^+$ is the one on the side of the normal vector.
  • Figure 7: Sector between given by $B_1$ and $B_2$ when the angle between them is smaller than $\pi$.
  • ...and 30 more figures

Theorems & Definitions (30)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 7
  • Proposition 9
  • Remark 10
  • Remark 12
  • Definition 13
  • Proposition 14
  • Remark 15
  • ...and 20 more