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

## Steve Boyer – Université du Québec à Montréal, Canada

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.