An atoroidal surface bundle over a surface

Consider a fibre bundle over the circle whose fibre is a closed surface of genus at least 2. William P. Thurston proved that the total space admits a hyperbolic metric if and only if the monodromy is pseudo-Anosov. The 4-dimensional analogue is a surface bundle over a surface, which is a smooth oriented fibre bundle whose fibre and base space are closed surfaces of genus at least 2. It is then natural to wonder whether there exists a surface bundle over a surface such that the total space admits a hyperbolic metric. This is still an open question. A weaker formulation of this question is the following: is there a surface bundle over a surface, such that the fundamental group of the total space contains no subgroups isomorphic to Z^2? Such a bundle is called atoroidal. The weaker formulation remained an open question for decades, until Autumn E. Kent and Christopher J. Leininger provided the first example of an atoroidal surface bundle over a surface. The key innovation in their work is the construction of a type-preserving, injective homomorphism from the figure-eight knot group to the mapping class group of the thrice-punctured torus. The aim of this thesis is to present the result of Kent and Leininger in detail. The construction combines the Birman exact sequence with results from algebraic and geometric topology.