Tree amalgamation of graphs and tessellations of the Cantor sphere

A general method is described which gives rise to highly symmetric tessellations of the Cantor sphere, i.e., the 2-sphere with the Cantor set removed and endowed with the hyperbolic geometry with constant negative curvature. These tessellations correspond to almost vertex-transitive planar graphs with infinitely many ends. Their isometry groups have infinitely many ends and are free products with amalgamation of other planar groups, possibly one or two-ended or finite. It is conjectured that all vertex-transitive tessellations of the Cantor sphere can be obtained in this way. gh our amalgamation construction is rather simple, it gives rise to some extraordinary examples with properties that are far beyond expected. For example, for every integer k, there exists a k-connected vertex-transitive planar graph such that each vertex of this graph lies on at least k infinite faces. These examples disprove a conjecture of Bonnington and Watkins that there are no 5-connected vertex-transitive planar graphs with infinite faces. This also disproves another conjecture that in a 4-connected vertex-transitive planar graph each vertex lies on the boundary of at most one infinite face. Further examples give rise to counterexamples of some other conjectures of similar flavor.