Examples of relatively hyperbolic groups. (Q1851081)

The author uses ideas on weak hyperbolicity and relative hyperbolicity which originate in work of Gromov. He uses the following definition of weak hyperbolicity that has been formulated by \textit{B. Farb} [in Geom. Funct. Anal. 8, No. 5, 810-840 (1998; Zbl 0985.20027)]. One starts with the Cayley graph \(\Gamma\) of a group \(G\) equipped with a finite generating system, and defines a new graph \(\widehat\Gamma=\widehat\Gamma(H)\) from \(\Gamma\) by adding a vertex \(v(gH)\) for each coset \(gH\) of \(H\) and an edge of length \(1/2\) from each element of \(gH\) to the vertex \(v(gH)\). The graph \(\widehat\Gamma\) is equipped with its natural path-metric. The group \(\Gamma\) is said to be `weakly hyperbolic with respect to \(H\)' if \(\widehat\Gamma\) is a negatively curved space in the sense of Gromov. The result of this paper is the following Theorem: Let \(H\) be a finitely presented group. Then \(H\) is a subgroup of a finitely presented group \(G\) which retracts onto \(H\) so that \(G\) is weakly hyperbolic with respect to \(H\). The proof uses ideas of relative hyperbolization of polyhedra originating in \textit{M. Gromov}'s paper [Essays in group theory, Publ., Math. Sci. Res. Inst. 8, 75-263 (1987; Zbl 0634.20015)], as well as developments of these ideas by \textit{M. W. Davis} and \textit{T. Januszkiewicz} [in J. Differ. Geom. 34, No. 2, 347-386 (1991; Zbl 0723.57017)]. The paper also contains open problems.