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 Lspace - 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.

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