  For each Cantor set C in $R^{3}$, all points of which have bounded local
genus, we show that there are infinitely many inequivalent Cantor sets in
$R^{3}$ with complement having the same fundamental group as the complement of
C. This answers a question from Open Problems in Topology and has as an
application a simple construction of nonhomeomorphic open 3-manifolds with the
same fundamental group. The main techniques used are analysis of local genus of
points of Cantor sets, a construction for producing rigid Cantor sets with
simply connected complement, and manifold decomposition theory. The results
presented give an argument that for certain groups G, there are uncountably
many nonhomeomorphic open 3-manifolds with fundamental group G.
