Negatively curved graph and planar metrics with applications to type

P. Bowers

Every proper (Gromov) negatively curved metric space whose boundary contains a nontrivial continuum admits a (2,C)-quasi-isometric embedding of a uniform binary tree. We apply this result to various "type" problems, including questions of recurrence or transience of random walks on graphs and questions of parabolicity or hyperbolicity of circle packings. Though our graphs and circle packings are locally finite, there is no assumption of bounded degree, nor of any isoperimetric condition.