I'm working through "Groups, Graphs and Trees" by John Meier.
In Chapter 5, he states that in a Cayley Graph, $\Gamma$ (or indeed any graph), the ball of radius n centred at the vertex v, $\mathcal{B}(v,n)$, is the subgraph formed as the union of all paths in $\Gamma$ of length $\le n$ that start at the vertex $v$.
It is then stated that a Cayley graph Γ is constructible if given $n \in N$ one can construct $\mathcal{B}(e,n)$ in a finite amount of time.
My question is what is an example of a non-constructible graph?
In particular, it is implied that there are finitely generated groups which have non-constructible Cayley graphs. I cannot comprehend what such a group looks like.
To construct the Cayley graph you need to be able to solve the word problem in the group, so any group with unsolvable word problem is an example.
In fact the converse is also true - if the group has solvable word problem then you can construct the Cayley graph.