Fundamental Group of a Cayley Graph

[Monobrow]

Suppose we have a group with presentation G = <A|R> i.e G is the quotient of the free group F(A) on A by the normal closure <<A>> of some subset A of F(A). Is it true that that fundamental group of the Cayley graph of G (with respect to the generating set A) will be isomorphic to the subgroup <<A>> in F(A)? It seems to me that this should be true (and it agrees with the facts that: a subgroup of a free group is free and the fundamental group of a graph is free) but I can't find this theorem stated anywhere...

 

[Bacle2]

Check the Reidemesiter-Schreier method; I am kind of running now, but see, e.g:

http://books.google.com/books?id=5B...nepage&q=reidemeister schreier method&f=false

 

[Monobrow]

Thanks for the reference, I'll take a look when I have the chance. I was thinking that I may have a method to prove what I was asking. The Cayley graph of F(A) is a tree T. Now F(A) acts freely and properly discontinuously on T, and thus any subgroup must also act freely and properly discontinuously (this is easy to show). So we have a free properly discontinuous action of <<S>> on T. Now if it were true that T/<<S>> were the Cayley graph of G then we would be done, since T is the universal cover of T/<<S>>. Again, I am convinced that this statement is true but I have haven't seen it stated in ANY reference which I find strange.

 

[Monobrow]

I've just noticed that I wrote <<A>> when I meant <<R>> in the first post and <<S>> when I meant <<R>> in my second post, sorry about that!

 

[blue_raver22]

I can confirm that your statement is indeed true. The fundamental group of the Cayley graph of a group G, with respect to a generating set A, is isomorphic to the subgroup <<A>> in F(A). This is known as the Stallings theorem, which states that the fundamental group of a graph of groups is isomorphic to the amalgamated free product of the fundamental groups of the vertex groups. In this case, the vertex groups are the free group F(A) and the amalgamated subgroup is <<A>>, which is the normal closure of A in F(A).

This theorem has important implications in group theory and topology, as it allows us to understand the fundamental group of a complicated space by breaking it down into simpler pieces. It also helps us to understand the structure of groups and their subgroups.

While the theorem may not be explicitly stated in all sources, it is a well-known result in the field of algebraic topology and is often used in the study of graphs and groups. I recommend consulting with a mathematician or a topology textbook for further information and examples of this theorem in action.