By Benjamin Fine, Gerhard Rosenberger
A survey of one-relator items of cyclics or teams with a unmarried defining relation, extending the algebraic research of Fuchsian teams to the extra common context of one-relator items and similar crew theoretical concerns. It presents a self-contained account of definite typical generalizations of discrete teams.
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Extra info for Algebraic Generalizations of Discrete Grs: A Path to Combinatorial Gr Theory Through One-Relator Products
For any generator g, let g~ = t~gt -~, i E Z.
1 One-Relator Groups and the Freiheitssatz As mentioned in the last chapter the theory of one-relator groups has always been of central importance in combinatorial group theory. The reasons for this are varied and arise from both topological and complex function theoretic as well as algebraic considerations. From a topological viewpoint, the class of surface groups, that is the fundamental groups of finite genus compactsurfaces, fall into the class of one-relator groups. These surface groups in turn, are part of the collection of Fuchsian groups, which are of great interest in complexfunction theory.
Now suppose a group G acts on an R-tree T. We say g ~ G is an inversion if g leaves a segment invariant but g has no fixed points. As before G acts freely on an JR-tree if there are no fixed points. An R-free group is a group which acts freely and without inversions on an R-tree. Clearly free groups are R-free. Further free abelian groups and all orientable surface groups of any genus as well as all non-orientable surface groups of genus _> 4 also act freely on R-trees. In fact, in a sense these are the only finitely generated examples.
Algebraic Generalizations of Discrete Grs: A Path to Combinatorial Gr Theory Through One-Relator Products by Benjamin Fine, Gerhard Rosenberger