The fundamental group of Cayley graph 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.
 A: As others have said, given a group $G$, its Cayley graph $\Gamma$ need not satisfy $\pi_1(\Gamma) \cong G$.  However, there is a notion of a Cayley complex $X_G$ which does satisfy $\pi_1(X_G) \cong G$.  Cf. Hatcher pages 52 and 77, which I quote below

Choose a presentation $G = \langle g_\alpha \mid r_\beta \rangle$.  This exists since every group is a quotient of a free group, so the $g_\alpha$ can be taken to be the generators of this free group with the $r_\beta$ generators of the kernel of the map from the free group to G.  Now construct $X_G$ from $\bigvee_\alpha S^1_\alpha$ by attaching 2-cells $e^2_\beta$ by the loops specified by the words $r_\beta$.

This produces a cell complex of dimension 2, which is just a graph with some discs glued on around loops.  It's really fun to try this with some simple examples.  Try it with $G = \mathbb{Z}/2\mathbb{Z} = \langle a \mid a^2 = 1\rangle$: you should get that $X_G$ is $\mathbb{P}^2(\mathbb{R})$, the real projective plane!
If instead you take the Cayley graph of $G$ and glue on 2-cells for relations, you'll get  the universal covering space of $X_G$.  This is discussed in detail on p. 77.
A: Keep in mind that, in general, there is no canonical Cayley graph associated to a group as its construction depends on a choice of generators and relations. Apart from this problem, it is also not true that $G$ is isomorphic to the fundamental group of a Cayley graph of $G$ in most cases because the fundamental group of a connected graph is either trivial (in the case of a tree) or free. And, if $G$ is free then it has a Cayley graph which is an (infinite) tree and so has a trivial associated fundamental group.
The only case I can think of is the trivial Cayley graph of the trivial group which happens to have a trivial fundamental group (although there are other Cayley graphs of the trivial group which are not contractible, for instance the presentation $1=\langle a\mid a\rangle$ has a Cayley graph homeomorphic to a circle hence free fundamental group on one generator).
A: There is a relation between these groups in general, but a pretty weak one, via the short exact sequence: 
$$
1\to F\to H\to G\to 1,
$$
where $F=\pi_1$ of a Cayley graph $\Gamma$ of the group $G$ and $H$ is the fundamental group of the graph $X=\Gamma/G$. The groups $F$ and $H$, of course are free.  In other words, the groups $F$ and $G$ are related as the kernel and cokernel of a short exact sequence. 
This relation comes from the covering map $\Gamma\to X$ for which $G$ is the group of covering transformations. 
A: 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. 
