Let $... \leftarrow G_i \leftarrow G_{i+1} \leftarrow ...$ be an inverse system of graphs (which might not be locally finite), morphisms are symplicial maps (we allow collapsing of edges to vertices).
We define a combinatorial inverse limit as follows:
- vertices are sequences ${x_i}$ such that each $x_i$ is a vertex for $G_i$,
- vertices ${x_i}$ and ${y_i}$ are connected iff for every $i$ we have: $x_i$ and $y_i$ are equal or are connected in $G_i$.
As a set, a combinatorial inverse limit is same as the standard inverse limit of topological spaces, but the topology is different. In particular, a combinatorial inverse limit of connected graphs might be disconnected.
This construction seems very natural and I wonder if there is anything known about that?
I want to study how Gromov's hyperbolicity behaves when one passes to the inverse limit. The main example is the inverse limit of graphs of homotopy classes of curves on a surface.
Thus far someone told me about book by Luis Ribes but its main focus is about profinite properties of such limit. (question was also asked on Math.SE about a month ago, though the only answer was one listed above)
