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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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