I am looking for examples of hyperbolic groups that have boundary homeomorphic to the 2-sphere, $S^2$. I would like an explicit presentation of such a group so that I can draw its Cayley graph and play around with it. I know some examples are fundamental groups of hyperbolic 3-manifolds and hyperbolic Coxeter groups, but it is quite difficult to find explicit presentations.
If someone could also explain how to prove that its boundary is in fact the 2-sphere, that would be quite helpful. Yes, this is related to Cannon's Conjecture.
The easiest way to explicitly draw the Cayley graph is to look at fundamental domains and take the dual graph.
Take a right-angled dodecahedron; you can think of its dual as a vertex with 12 edges going out. Attach a new right-angled dodecahedron to each of those faces. So far the graph we have is (part of) a valence-12 tree.
In the next step, we add on more right-angled dodecahedra, but now we 'complete the right-angles', so a single fundamental domain gets attached to two adjacent faces at once. This creates four-cycles in the graph. Continuing on in this way will easily get you the Cayley graph. This is by far the simplest hyperbolic Cayley graph with a 2-sphere boundary that I know of.