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We prove that every homomorphism from the fundamental group of a planar Peano continuum to the fundamental group of a planar or one-dimensional Peano continuum is induced by a continuous map up to conjugation. This is then used to provide a family of uncountable many planar Peano continua with pairwise non-isomorphic fundamental groups all of which are not homotopy equivalent to a one-dimensional space.of any one-dimensional Peano continuum.Proof. Let {U 1 , U 2 , · · · } be a countable set of disjoint open subsets of (0, 1)×(0, 1). In each U i we can find a subset X i such that X i ⊂ S and X i is homeomorphic to the wedge of i-closed intervals.ForĀ ⊂ N, let XĀ = i∈Ā X i . It is a trivial exercise to show that that XĀ is homeomorphic to XB if and only ifĀ =B.ForĀ ⊂ N, choose A ⊂ N such that K(S A ) = XĀ. The corollary then follows from Proposition 2.14.

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We define a local analogue to Gromov's loop division property which is use to give a sufficient condition for an asymptotic cone of a complete geodesic metric space to have uncountable fundamental group. As well, this property is used to understand the local topological structure of asymptotic cones of many groups currently in the literature

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