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Planarity of orbital Schreier graphs of free products of cyclic groups. (English) Zbl 1374.05111

Summary: Free products of finite number of finite groups admit faithful actions by finite automaton permutations. Each action of this kind gives rise to Schreier graphs on levels and orbital Schreier graphs. These graphs depend on the action and a fixed generation set of the group. For the free product of two nontrivial cyclic finite groups we construct an orbital Schreier graph such that its underlying graph is planar. The proof relies on the Kuratowski planarity criterion for infinite graphs and uses inductive limits of finite planar graphs.

MSC:

05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
05C10 Planar graphs; geometric and topological aspects of graph theory
20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
20D06 Simple groups: alternating groups and groups of Lie type
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