TITLE: The fundamental group of Cayley graph
QUESTION [5 upvotes]: Today I read a math book and find interested in 

Theorem. Every group has its graph representation.

And we call it cayley graph. Now we sort out the question

Firstly, if we have a group, then we can construct a graph.
   Second, if we have a graph, then we can induce a topological structure. So we have $\pi_1(G)$ by denoting the graph $G$.

My question is what's relationship between the origin group and $\pi_1(G)$. Are they equal? I need an answer which just need to clarify their vague relationship. Thanks.

REPLY [2 votes]: I would like to draw attention to the paper 
Ellis, G. "Computing group resolutions". J. Symbolic Comput. 38 (2004) 1077--1118.
and its  implementation in GAP as shown in Graham's web page Homological Algebra Programming. 
It is relevant to this question on Cayley graphs because for a class of groups $G$, in this case automatic ones, it starts with the Cayley graph of a presentation, and then constructs the universal cover $Y$ of a $K(G,1)$ together with a contracting homotopy of $Y$. In other words, given the construction of $Y$ up to level $n$ and a "partial contracting homotopy" up to that level, this information is used to construct the next level of the universal cover with the next level of contracting homotopy. The start is a maximal  tree in the Cayley graph! So the Cayley graph up to level $2$ as described in Hatcher's book referred to by SpamIAm is the $2$-skeleton of the universal cover $Y$.  The paper describes the relation with previous work. 
I like to express this construction by saying that the information up to level $n$ enables one to construct, and then refine, a "home for a contracting homotopy" at the next level.  This is a change from the traditional "killing kernels" method in homological algebra.