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On Planar Cayley graphs and Kleinian groups

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Figure

Figure 1: The path Q in the proof of Lemma 5.3 (dashed).
Figure 2: The standard Cayley graph of Z4 × Z2.
Figure 3: A non-planar Cayley graph ofgraph Z4 ×Z2. This graph is the complete bipartite K4,4: the vertices at the same horizontal/vertical level form the two partitionclasses.
Figure 4:The situation whenare pairwise disjoint. P, gP, g2P (displayed in green, if colour is shown)We show that this situation can be achieved by choosing Pappropriately.

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