§ Hyperbolic groups have solvable word problem
I have never seen an elementary account of this in a 'trust these facts, now
here is why hyperbolic groups have a solvable word problem'. I am writing
such an account for myself. It's an account for building intuition, so no
proofs will be provided except for the final theorem. All facts will be backed
by intuition. Since most of it is geometric, it's easy to convey intuition.
§ Graphs of groups, quasi-isometry.
We can convert any group into a graph by using the cayley graph of the group.
We characterize hyperbolic space as a space where we can build 'thin triangles'.
We also think of hyperbolic space as where geodesics from a given point
diverge (in terms of angle) exponentially fast.
The choice of generators for the cayley graph gives different graphs. We can
assign a unique geometric object by considering cayley graphs upto quasi
isometry. The cayley graph of a group with respect to different generating sets
are quasi-isometric. We can now try to study properties that are invariant
under quasi-isometry, since these are somehow 'represented faithfully by the geometry'.
- NOTE: I will consistently denote the inverse of by .
§ Hyperbolicity enters the picture
We now say that a graph is hyperbolic if the cayley graph of the group is
hyperbolic. We can show that hyperbolicity is preserved by quasi-isometry.
So this property does not depend on the generating set.
§ Isoperimetric Inequality
If we have a finitely presented group , and
is a word in the free group , if , we will have
This follows almost by definition. Since we have quotiented by we can have
elements of in between . We will need to have a since there's
nothing else to cancel off the .
§ Area of a word
Let . Let be a word in such that .
The area of the word is the minimal number of such components
we need to write it down. Formally:
I don't understand the geometric content of this definition. I asked
on mathoverflow .
§ Isopermetric function for a group
is a Dehn function or isoperimetric function
if the area of the word is upper bounded by . In some sense, the
length of the word is the perimeter to the area, and this gives us a form
of the isoperimetric inequality. Formally, is a Dehn function if for all
words such that , we have . depending
on the growth of , we say that has linear, quadratic, exponential etc.
§ Geometric content of Area
We define a map to be aninite, planar, oriented, connected and simply connected
simplicial2-complex (!). A map is a diagram over an alphabet iff every
edge has a label such that .
Hang on: what does it mean to invert an edge? I presume it
means to go backwards along an edge. So we assume the graph is directed, and we
have edges in both directions.
A Van Kampen diagram over a group is a diagram
over such that for all faces of , the label of the boundary of
is labelled by some . The area of such a diagram
is the number of faces.
§ Hyperbolic iff Linear Isopermetric Inequality is satisfied
A finitely presented group is hyperbolic if and of if its cayley grah
satisfies the linear isoperimetric inequality.
§ Deciding if elements are conjugate to each other
- If we can answer the question of whether two elements are conjugate to each other (does there exist a such that ), we can solve that an element is equal to the identity:
- Pick . Then if we have , then hence .
- If we can check that an element is equal to the identity, we can check for equality of elements. two elements are equal iff .
- So solving conjugacy automatically allows us to check of equality.
§ Proof that conjugacy is solvable for hyperbolic groups
Consider a solution to the problem of finding an such that .
We claim that due to the hyperbolicity of the space, such an cannot be