Left-orderings of groups and the topology of 3-manifolds

Many decades of work culminating in Perelman's proof of Thurston's geometrisation conjecture showed that a closed, connected, orientable, prime 3-dimensional manifold \(W\) is essentially determined by its fundamental group \(\pi_1(W)\). This group consists of classes of based loops in \( W\) and its multiplication corresponds to their concatenation. An important problem is to describe the topological and geometric properties of $W$ in terms of \(\pi_1(W)\). For instance, geometrisation implies that \(W\) admits a hyperbolic structure if and only if \(\pi_1(W)\) is infinite, freely indecomposable, and contains no \(\mathbb Z \oplus \mathbb Z\) subgroups. In this talk I will describe recent work which has determined a surprisingly strong correlation between the existence of a left-order on \(\pi_1(W)\) (a total order invariant under left multiplication) and the following two measures of largeness for \(W\): a) the existence of a co-oriented taut foliation on \(W\) - a special type of partition of \(W\) into surfaces which fit together locally like a deck of cards. b) the condition that \(W\) not be an L-space - an analytically defined condition representing the non-triviality of its Heegaard-Floer homology. I will introduce each of these notions, describe the results which connect them, and state a number of open problems and conjectures concerning their precise relationship.