Domination between non-Fuchsian representations and anti-de Sitter geometry

TL;DR

This paper investigates domination relations between non-Fuchsian surface group representations, linking them to anti-de Sitter 3-manifolds with singularities, and provides solutions for cases with branched harmonic immersions.

Contribution

It introduces a new domination problem for non-Fuchsian representations, solves it for branched harmonic immersions, and constructs large families of singular anti-de Sitter 3-manifolds.

Findings

Domination problem solvable for representations with branched harmonic immersions

Outside this case, the domination problem cannot always be solved

Dominating pairs lead to the construction of singular anti-de Sitter 3-manifolds

Abstract

Motivated by work of various authors on domination between surface group representations, harmonic maps, and $3$-dimensional anti-de Sitter geometry, we study a new domination problem between non-Fuchsian representations of closed surface groups. We solve the problem for representations that admit branched harmonic immersions, and we show that, outside of this case, the problem cannot always be solved. We then show that a dominating pair gives rise to an anti-de Sitter $3$-manifold with singularities, and we construct large families of branched anti-de Sitter $3$-manifolds.