## Sufficiently Rich Families of Planar Rings

###
J. W. Cannon, W. J. Floyd, and W. R. Parry

July 22, 1996

### Abstract

It has been conjectured that if $G$ is a negatively curved discrete group with
space at infinity $\partial G$ the 2-sphere, then $G$ has a properly
discontinuous, cocompact, isometric action on hyperbolic 3-space. Cannon and
Swenson reduced the conjecture to determining that a certain sequence of
coverings of $\partial G$ is conformal in the sense of Cannon's combinatorial
Riemann mapping theorem. In this paper it is proved that, in this setting, the
two axioms of conformality can be replaced by a single axiom which is implied by
each of them.
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