Definition N34.1 (Free group). The free group on generators consists of the words
It is the “largest” group generated by elements: all others are to quotients of .
If is generated by , we define a homomorphism by , i.e.
Definition N34.2 (Finitely presented). is finitely presented if the kernel of is the smallest normal subgroup of containing some finite subset .
Example N34.3.
In general
with the composite equal to . The form a complete set of relations for the group given by the .
Cayley graphs
Definition N34.4 (Cayley graph). The Cayley graph of with generators : the vertices are the elements of the group, with an edge when .
Example N34.5 ( with generator ).
Example N34.6 ( with ,). Edges are multiplication by a generator or its inverse; since are their own inverses, each edge is an arrow in both directions, drawn as a plain edge.
Here ,, and .
RAHHH I LOVE CAYLEY GRAPHS!!!
Geometric group theory
… (unfinished in the notes).
(Margin: manipulating the geometric structure of Cayley graphs with forced dimensionality?)
acts on its Cayley graph by left multiplication, transitively and with no fixed points (left multiplication by fixes no vertex).
Word length …
Algorithmic and group complexity: this is akin to saying how complexity increases with added [generators/relations?], polynomially or exponentially. (P = NP relation? I wonder what techniques are used.)