Setting the stage: Low-dimensional topology and the fundamental group.
Article: Aschenbrenner, Friedl, Wilton, 3-manifold.groups (2012) [public: ArXiv]
In 3-manifold topology, the fundamental group captures an immense amount of information about a manifold, including irreducibility, fibering over the circle, fibering by circles (Seifert fibering), containing essential tori, etc. When a new group-theoretic concept comes along, a natural question to ask is: what topological proerty does it reflect?

(Left) orderings on groups.
Text: Clay and Rolfsen, Ordered groups and Topology (2016)
Article: Sunic, .Orders on free groups induced by oriented words (2013) [public:ArXiv]
Basic concepts, positive cones; examples: Homeo(R), free group (Ping Pong Lemma); characterizations of orderability; local indicability, Kaplansky's Conjecture; embedding LO-groups in Homeo(R).

Knot theory, 3-manifolds, and left orderings.
Text: Clay and Rolfsen, Ordered groups and Topology (2016)
Text: Stillwell, Classical Topology and Combinatorial Group Theory (1980)
Knot groups, Wirtinger presentations, local indicability. Ordering 3-manifold groups, left-orderable quotients, convex subgroups. Dehn surgery/Dehn filling, Heegaard splittings, surgery description of a 3-manifold.

Non-left-orderable groups.
Article: Clay and Watson, Left-orderable fundamental groups and Dehn surgery (2012)
Article: Boyer, Gordon, and Watson, On L-spaces and left-orderable fundamental groups (2011)
Dehn fillings of torus knot exteriors (Dehn surgery on a torus knot). Branched coverings, the non-LO-bility of double branched covers over an alternating knot.

Foliations.
Text: Calegari, Foliations and the Geometry of 3-manifolds (2007)
Article: Lickorish, A foliation for 3-manifolds (1965)
Article: Calegari and Dunfield, Laminations and groups of homeomorphisms of the circle (2002)
Definitions, constructions, all 3-manifolds foliate. Taut/Reebless foliations, the space of leaves, R-covered foliations. Foliated Seifert-fibered spaces; R-covered implies left-orderable fundamental group, tautly foliated implies virtually left-orderable.
Big question: are tautly foliated and left-orderable (essentially) the same? Experimental evidence/odd coincidences suggest yes!

Heegaard Floer homology.
Article: Ozsvath and Szabo, An Introduction to Heegaard Floer Homology
Article: Ozsvath and Szabo, Heegaard Floer homology and alternating knots (2003)
Article: Sahamie, Introduction to the basics of Heegaard Floer homology (2013)
Motivation: the seeming strong connections between LO, tautly foliated, and `non-simple' HF homology.
Pointed Heegaard diagrams, equivalence, (tori in) symmetric product; Whitney disks, Spin^c structures, Morse functions, the differential for HF-hat. Knot Floer homology, categorification of the Alexander polynomial, detection of knot genus. Knot surgery as 2-handle addition, 3-manifold triads, the surgery exact triangle. L-spaces, L-spaces from Dehn surgery on knots, L-spaces from double branched covers of alternating knots. L-spaces have no taut foliations!

Ordering braid groups.
Article: Fenn, Greene, Rolfsen, Rourke, and Wiest, Ordering the braid groups (1999)
Braid groups, Artin presentation, mapping class groups, arc diagrams; Dehornoy ordering, i-positive words.