Toledo invariant of lattices in SU(2,1) via symmetric square

TL;DR

This paper investigates the quaternionic Toledo invariant for lattices in SU(2,1), constructing multiple representations to reveal the structure of the character variety and exploring lifts to period domains.

Contribution

It introduces new representations to demonstrate multiple components in the character variety of complex hyperbolic lattices and studies holomorphic lifts to period domains.

Findings

Character variety has at least four distinct components.

Constructed four explicit representations.

Established existence of holomorphic horizontal lifts.

Abstract

In this paper, we address the issue of quaternionic Toledo invariant to study the character variety of two dimensional complex hyperbolic uniform lattices into $SU(n,2)$. We construct four distinct representations to prove that the character variety contains at least four distinct components. We also address the existence of holomorphic horizontal lift to various period domains of $SU(n,2)$.