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

TL;DR

This paper introduces a Riemannian metric on the space of maximal surface group representations into SO(2,n+1), revealing geometric structures and properties of these moduli spaces, especially in low and higher dimensions.

Contribution

It constructs a non-degenerate scalar product on cohomology for maximal representations, analyzing the metric's properties and geometric features in various cases.

Findings

The scalar product induces a Riemannian metric on maximal representations.

The metric is compatible with orbifold structures in certain cases.

The space of representations into SO(2,3) is shown to be totally geodesic.

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 $n\ge 3$.