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On subgroups of finite complexity in groups acting on trees. (English) Zbl 1107.20021
Suppose that $$G$$ is a group acting on a tree $$X$$. Given a subgroup $$H$$ of $$G$$, the author studies the existence of invariants of $$H$$, uniquely determined by the action of $$H$$ on $$X$$. For this he describes some properties of subgroups of finite complexity, an extension of the Schreier index formula for the rank of subgroups of finite index in free groups, to the case of subgroups of finite index in amalgamated free products. He also shows that all isomorphic subgroups of finite index in a finitely presented group with infinitely many ends have the same index, and that fixed subgroups of symmetric endomorphisms of fundamental groups of graphs of groups whose edge groups are all finite, are tame. Finally, the chains of subgroups of finite complexity are described.

MSC:
 20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations 20E07 Subgroup theorems; subgroup growth 20E08 Groups acting on trees 20F05 Generators, relations, and presentations of groups 20F65 Geometric group theory
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