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Riemannian geometry of maximal surface group representations acting on pseudo-hyperbolic space

Nicholas Rungi

Abstract

For any maximal surface group representation into $\mathrm{SO}_0(2,n+1)$, we introduce a non-degenerate scalar product on the the first cohomology group of the surface with values in the associated flat bundle. In particular, it gives rise to a non-degenerate Riemannian metric on the smooth locus of the subset consisting of maximal representations inside the character variety. In the case $n=2$, we carefully study the properties of the Riemannian metric on the maximal connected components, proving that it is compatible with the orbifold structure and finding some totally geodesic sub-varieties. Then, in the general case, we explain when a representation with Zariski closure contained in $\mathrm{SO}_0(2,3)$ represents a smooth or orbifold point in the maximal $\mathrm{SO}_0(2,n+1)$-character variety and we show that the associated space is totally geodesic for any $n\ge 3$.

Riemannian geometry of maximal surface group representations acting on pseudo-hyperbolic space

Abstract

For any maximal surface group representation into , we introduce a non-degenerate scalar product on the the first cohomology group of the surface with values in the associated flat bundle. In particular, it gives rise to a non-degenerate Riemannian metric on the smooth locus of the subset consisting of maximal representations inside the character variety. In the case , we carefully study the properties of the Riemannian metric on the maximal connected components, proving that it is compatible with the orbifold structure and finding some totally geodesic sub-varieties. Then, in the general case, we explain when a representation with Zariski closure contained in represents a smooth or orbifold point in the maximal -character variety and we show that the associated space is totally geodesic for any .
Paper Structure (16 sections, 35 theorems, 109 equations)

This paper contains 16 sections, 35 theorems, 109 equations.

Table of Contents

  1. Introduction
  2. Background materials
  3. Pseudo-hyperbolic space and maximal space-like surfaces
  4. The character variety and maximal surface group representations
  5. Nonabelian Hodge correspondence
  6. A Riemannian metric for $\mathrm{SO}_0(2,n+1)$ maximal representations
  7. Definition of the metric
  8. Relation with maximal space-like surfaces in H^2,n
  9. Sub-varieties for $n=2$
  10. Maximal connected components
  11. Orbifold singularities
  12. Totally geodesic sub-varieties and the Fuchsian locus
  13. A note about the Hitchin component
  14. Inclusions for $n\ge 3$
  15. The case $sw_1\neq 0$
  16. ...and 1 more sections

Key Result

Theorem 1

A For any maximal representation $\rho:\pi_1(\Sigma)\to\mathrm{SO}_0(2,n+1)$ which is also a smooth point in $\mathfrak{R}_{2,n+1}^{\text{max}}(\Sigma)$ and for any $n\ge 2$, there exists a scalar product $\mathbf{g}_\rho$ on the Zariski tangent space depending on the unique $\rho$-equivariant maxim

Theorems & Definitions (59)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3: goldman1984symplectic
  • Remark 2.4
  • Definition 2.5
  • ...and 49 more