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A non-trivial family of trivial bundles with complex hyperbolic structure

Hugo C. Botós, Felipe A. Franco

Abstract

In $\mathrm{PU}(2,1)$, the group of holomorphic isometries of the complex hyperbolic plane, we study the space of involutions $R_1, R_2, R_3, R_4, R_5$ satisfying $R_5R_4R_3R_2R_1=1$, where $R_1$ is a reflection in a complex geodesic and the other $R_i$'s are reflections in points of the complex hyperbolic plane. We show that this space modulo $\mathrm{PU}(2,1)$-conjugation is bending-connected and has dimension $4$. Using this, we construct a $4$-dimensional bending-connected family of complex hyperbolic structures on a disc orbibundle with vanishing Euler number over the sphere with $5$ cone points of angle $π$. Bending-connectedness here means that we can naturally deform the geometric structure, like Dehn twists in Teichmüller theory. Additionally, finding complex hyperbolic disc orbibundles with vanishing Euler number is a hard problem, originally conjectured by W. Goldman and Y. Eliashberg and solved by S. Anan'in and N. Gusevskii, and we produce a simpler and more straightforward construction for them.

A non-trivial family of trivial bundles with complex hyperbolic structure

Abstract

In , the group of holomorphic isometries of the complex hyperbolic plane, we study the space of involutions satisfying , where is a reflection in a complex geodesic and the other 's are reflections in points of the complex hyperbolic plane. We show that this space modulo -conjugation is bending-connected and has dimension . Using this, we construct a -dimensional bending-connected family of complex hyperbolic structures on a disc orbibundle with vanishing Euler number over the sphere with cone points of angle . Bending-connectedness here means that we can naturally deform the geometric structure, like Dehn twists in Teichmüller theory. Additionally, finding complex hyperbolic disc orbibundles with vanishing Euler number is a hard problem, originally conjectured by W. Goldman and Y. Eliashberg and solved by S. Anan'in and N. Gusevskii, and we produce a simpler and more straightforward construction for them.

Paper Structure

This paper contains 28 sections, 17 theorems, 134 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hyperbolic spaces
  4. The complex hyperbolic plane
  5. Geodesics
  6. Bisectors
  7. Triangle of bisectors
  8. Isometries of the complex hyperbolic plane
  9. Holonomy on a triangle of bisectors
  10. Representations of the hyperelliptic group
  11. PU(1,1)-Representations of the hyperelliptic group
  12. PU(2,1)-Representations of the hyperelliptic group
  13. Bendings and bending-deformations of representations
  14. The dimension of the representation space
  15. Decomposing isometries as the product of three reflections
  16. ...and 13 more sections

Key Result

Lemma 2.3

Goldman1999 Let $L_1,L_2$ be hyperbolic lines and let $p_1,p_2\in{\mathrm E}\,V$ be their polar points, respectively. Then $L_1$ and $L_2$ are ultraparallel, asymptotic, concurrent if and only if $\mathop{\mathrm{ta}}(p_1,p_2)>1$, $\mathop{\mathrm{ta}}(p_1,p_2)=1$, $\mathop{\mathrm{ta}}(p_1,p_2)<1$,

Figures (13)

  • Figure 1: A meridian $R\coloneq \mathbb P(W+\mathbb{R}\varepsilon f) \cap \overline{\mathbb{H}^2_{\mathbb{C}}}$ with three meridional curves: the geodesic $G$ in red is the meridional curve through $\xi_1$; the hypercycle of $G$ in purple is the meridional curve through $\xi_2$; the segment of $\partial R$ in blue containing $\xi_3$ is the meridional curve through $\xi_3$.
  • Figure 2: Surface $\mathcal{S}_{\pmb\sigma,\tau}$ in $\mathbb{R}^3$ obtained by taking $\pmb\sigma\coloneq (+,-,-)$ and $\tau\coloneq -2.22 - 3.84515\, i$. Vertical (resp. horizontal) lines obtained by keeping $s_1$ (resp. $s_2$) constant are marked.
  • Figure 3: Goldman's deltoid $\Delta$ and the line $\ell_{-1}$ tangent to $\Delta$
  • Figure 4: The construction of the quadrangle of bisectors and its decomposition as two transversally adjacent triangles
  • Figure 5: The face-pairing of the quadrangle of bisectors
  • ...and 8 more figures

Theorems & Definitions (48)

  • Remark
  • Remark
  • Remark
  • Lemma 2.3
  • Remark
  • Remark
  • Remark
  • Remark
  • Remark
  • Remark
  • ...and 38 more