group theory

[gusl]

Joel Friedman, who is my de jure supervisor at UBC, recently proved the Strengthened Hanna Neumann conjecture. (He is completely separated from my actual research here, which is roughly in Biostatistics)

This stuff sounds pretty neat, so I just spent about an hour reading up on group theory on Wikipedia, but as one might imagine, it got pretty tangential.

Things I understand:
Free group, e.g. homotopy groups where each generator loops around a different donut-hole.
Cayley graph, which makes nice visualizations of group structure
Word metric, i.e. the distance between two elements in the Cayley graph (I imagine this can be generalized by making edge weights a function of the corresponding element, rather than a constant)
Free abelian group, e.g. Z^2 under addition
Word problem for groups, though I don't understand how it can be unsolvable: is it by encoding the Halting problem?
Presentation of a group (but how is it related to group representations?)
Absolute presentation of a group
Commutator


Things I'd like to understand more clearly:
Automatic group
Heisenberg's uncertainty principle is related to a nonzero commutator