Abstract 

A total linear order on a non-trivial group G is a left-order if it’s invariant under group left-multiplication. A result of Boyer, Rolfsen and Wiest shows that a $3$-manifold group has a left-order if and only if it admits a non-trivial representation into Homeo_+(S^1) with zero Euler class. Foliations, laminations and flows on $3$-manifolds often give rise to natural non-trivial Homeo_+(S^1)-representations of the fundamental groups, which have proven to be extremely useful in studying the left-orderability of $3$-manifold groups.

In this talk, we will present a recipe of stir frying these Homeo_+(S^1)-representations. Our operation generalizes a previously known ``flipping'' operation introduced by Calegari and Dunfield. As a consequence, we constructed a surprisingly large number of new Homeo_+(S^1)-representations of the link groups. We then use these newly obtained representations to prove the left-orderablity of cyclic branched covers of links associated with any epimorphisms to Z_n. This is joint work with Steve Boyer and Cameron Gordon.