What is Geometric Group Theory?

We ask what is the symmetry (isometry) group is. For instance, consider a triangle:

We say isometry to mean "distance preserving". The various isometries are:

• Rotate 60 deg. CCW
• Rotate 120 deg. CCW
• Do nothing
• Flip across a
• Flip across b
• Flip across c
  All of these actions form a group. And we generalize to many shapes with symmetries.

Various isometries are:

• Rotations
• Reflections
• Glide Reflections
• ...

Geodesics (shortest paths)

We know in ${\mathbb{R}}^2$ that the shortest path between two points in a line. But if you asked that on a sphere? You'd have to travel along the arc of the circle, going in the straight line in perspective to that circle.

Another way to think of it is that we are "cutting" the sphere into two along that line of travel.

The Free Group $F_r$

For instance $F_3=<x_1,x_2,x_3>$ as well as their inverse. Inverses "multiplied" with their normal element cancel to the identity. It's just strings of elements and their inverses:

Some transformations are "structure preserving". You can think of them as the commutativity of a map with multiplication (ie: $f(x\ast c)=c\ast f(x)$). Sometimes we denote this at $\text{Aut}(F_r)$.

We define $\varphi$ as the images of their generators:

Marked Metric Graphs

We can denote marked, metric graphs of some group like $F_3$ via:

We can:

• Change lengths on edges