Various generalizations and deformations of $PSL(2,\mathbb{R})$ surface group representations and their Higgs bundles

Abstract

Recall that the group $PSL(2,\mathbb{R})$ is isomorphic to $PSp(2,\mathbb{R}),S{O}_0(1,2)$ and $PU(1,1).$ The goal of this paper is to examine the various ways in which Fuchsian representations of the fundamental group of a closed surface of genus $g$ into $PSL(2,\mathbb{R})$ and their associated Higgs bundles generalize to the higher rank groups $PSL(n,\mathbb{R}),PSp(2n,\mathbb{R}),S{O}_0(2,n),S{O}_0(n,n+1)$ and $PU(n,n)$. For the $S{O}_0(n,n+1)$-character variety, we parameterize $n(2g-2)$ new connected components as the total space of vector bundles over appropriate symmetric powers of the surface and study how these components deform in the $S{O}_0(n,n+2)$-character variety. This generalizes results of Hitchin for $PSL(2,\mathbb{R})$.

20 pages, written for Hitchin's 70th birthday proceedings

References in corpus (1)

• Topological Invariants of Anosov Representations