Relatively hyperbolic groups.

*(English)*Zbl 0985.20027From the introduction: We introduce a theory of relatively hyperbolic groups. Fundamental groups of complex (resp. quaternionic, Cayley) hyperbolic manifolds with cusps are examples of groups which are not word-hyperbolic (or even automatic), and, indeed, which lie outside many of the techniques in geometric group theory. The reason these groups differ so much from the real hyperbolic case, is that in the real hyperbolic case the geometry of the group turns out to be nonpositively curved. This is due essentially to the fact that the cusp groups are Abelian. The fundamental groups of complex (resp. quaternionic, Cayley) hyperbolic manifolds with cusps do not exhibit nonpositively curved geometry. Instead they combine a nontrivial mix of both negatively curved and nilpotent geometry.

The techniques introduced in this paper are meant to provide some machinery for dealing with groups which exhibit more than one type of geometric behavior. Our methods pick out and exploit aspects of negative curvature in a group \(\Gamma\), when \(\Gamma\) itself is not a word-hyperbolic group. We emphasize that we do not simply consider coset graphs, which is the naive approach (and does not work for the motivating examples).

Here is a rough idea of the definitions involved; for details see §3. Although the general theory applies to groups which are hyperbolic relative to a finite set of subgroups, we begin with the case of only one subgroup. Let \(H\) be a finitely-generated subgroup of a finitely-generated group \(G\). For any Cayley graph \(\Gamma\) of \(G\) we form a quotient graph \(\widehat\Gamma\) of \(\Gamma\) by identifying, for each \(g\in G\), all vertices of \(\Gamma\) corresponding to elements lying in the left coset \(gH\). Definition (Relatively hyperbolic group). The group \(G\) is hyperbolic relative to \(H\) if \(\widehat\Gamma\) is a negatively curved metric space.

We also isolate a property that can be satisfied by a pair \((G,H)\), where \(G\) is hyperbolic relative to \(H\): the Bounded Coset Penetration property (or BCP property for short). The BCP property strengthens the notion of malnormal subgroup, and will allow us to conclude information about \(G\) from corresponding information about \(H\).

The techniques introduced in this paper are meant to provide some machinery for dealing with groups which exhibit more than one type of geometric behavior. Our methods pick out and exploit aspects of negative curvature in a group \(\Gamma\), when \(\Gamma\) itself is not a word-hyperbolic group. We emphasize that we do not simply consider coset graphs, which is the naive approach (and does not work for the motivating examples).

Here is a rough idea of the definitions involved; for details see §3. Although the general theory applies to groups which are hyperbolic relative to a finite set of subgroups, we begin with the case of only one subgroup. Let \(H\) be a finitely-generated subgroup of a finitely-generated group \(G\). For any Cayley graph \(\Gamma\) of \(G\) we form a quotient graph \(\widehat\Gamma\) of \(\Gamma\) by identifying, for each \(g\in G\), all vertices of \(\Gamma\) corresponding to elements lying in the left coset \(gH\). Definition (Relatively hyperbolic group). The group \(G\) is hyperbolic relative to \(H\) if \(\widehat\Gamma\) is a negatively curved metric space.

We also isolate a property that can be satisfied by a pair \((G,H)\), where \(G\) is hyperbolic relative to \(H\): the Bounded Coset Penetration property (or BCP property for short). The BCP property strengthens the notion of malnormal subgroup, and will allow us to conclude information about \(G\) from corresponding information about \(H\).

##### MSC:

20F67 | Hyperbolic groups and nonpositively curved groups |

57M07 | Topological methods in group theory |

57M05 | Fundamental group, presentations, free differential calculus |

20F05 | Generators, relations, and presentations of groups |

20E06 | Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations |

20F10 | Word problems, other decision problems, connections with logic and automata (group-theoretic aspects) |