On the Varieties of Representations into GL(4;C): The Trefoil Knot

Abstract

Following the foundational work of Thurston, Culler and Shalen, the varieties of representations and characters of 3-manifold groups into SL(2,C) have been extensively studied, as both reflect geometric and topological properties of 3-manifolds. However, much less is known about the representations of 3-manifold groups into other Lie groups, notably SL(n,C) with n ≥3. This thesis investigates the representations of the fundamental group of the trefoil knot complement (denoted Γ) into the group GL(4,C). In particular, it studies the various types of representations of Γ into SL(2,C), SL(3,C), SL(4C) and GL(4, C). The main results describe character varieties of various types of such representations, imposing various conditions on reducibility. We determine all representations up to conjugacy, using results of Fricke for SL(2,C) and Lawton for SL(3,C). Parametrisations and reducibility criteria of some types of representations are also given. This is then extended to GL(4,C) and some geometric properties are studied.