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The relative hyperbolicity of one-relator relative presentations. (English) Zbl 1219.20030
The author continues Klyachko’s study of groups defined by one-relator relative presentations $$\widetilde G=\langle G,t\mid w^k\rangle$$ where $$w=g_1t^{\varepsilon_1}\cdots g_nt^{\varepsilon_n}$$ ($$\varepsilon_i=\pm 1$$) is unimodular (i.e. $$\sum_{i=1}^n\varepsilon_i=1$$) and $$G$$ is torsion-free. Here it is shown that if $$k\geq 2$$ then $$\widetilde G$$ is relatively hyperbolic with respect to $$G$$. The proof involves Howie diagrams, Klyachko’s method of multiple motions on diagrams, and a version of his car crash lemma. Corollaries give that $$\widetilde G$$ is SQ-universal when $$G$$ is non-trivial and that the word problem is solvable in $$\widetilde G$$ if it is solvable in $$G$$.

##### MSC:
 20F67 Hyperbolic groups and nonpositively curved groups 20F05 Generators, relations, and presentations of groups 20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects) 20F06 Cancellation theory of groups; application of van Kampen diagrams 20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations 20F70 Algebraic geometry over groups; equations over groups
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##### References:
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