TCS oriented refs/survey on group theoretic word problem
One family of results not mentioned in the excellent references of @Marzio's answer is relations between the isoperimetric (Dehn) function of a group $G$ and the nondeterministic time complexity of the word problem in $G$. For example:


For finitely generated groups $G$, $WP(G) \in \mathsf{NP}$ if and only if $G$ can be embedded in a finitely presented group that has polynomial Dehn function. (In fact, if $WP(G)$ can be solved in $\mathsf{NTIME}(T(n))$, then $G$ can be embedded in a group with isoperimetric function $\leq n^2 T(n^2)^4$.) (Birget-Ol'shanskii-Rips-Sapir)
There are individual groups whose word problem is $\mathsf{NP}$-complete (corollary to the preceding), and similarly groups whose word problem is $\mathsf{coNP}$-complete (Birget, also available on the arXiv).
The collection of functions $\mathcal{D}_4$ that are Dehn functions $\geq n^4$ of finitely presented groups are closely related to the collection $\mathcal{T}_4$ of time functions $\geq n^4$ of nondeterministic Turing machines. Specifically, $\mathcal{T}^4 \subseteq \mathcal{D}_4 \subseteq \mathcal{T}_4$, where $\mathcal{T}^4$ is the set of super-additive functions that are fourth powers of time functions of nondeterministic TMs. (Sapir-Birget-Rips)


See this survey by Sapir for more.