Relatively Hyperbolic

In mathematics, the concept of a relatively hyperbolic group is an important generalization of the geometric group theory concept of a hyperbolic group. The motivating examples of relatively hyperbolic groups are the fundamental groups of complete noncompact hyperbolic manifolds of finite volume.

Intuitive definition

A group ''G'' is relatively hyperbolic with respect to a subgroup ''H'' if, after contracting the Cayley graph of ''G'' along ''H''- cosets, the resulting graph equipped with the usual graph metric becomes a δ-hyperbolic space and, moreover, it satisfies a technical condition which implies that quasi-geodesics with common endpoints travel through approximately the same collection of cosets and enter and exit these cosets in approximately the same place.

Formal definition

Given a finitely generated group ''G'' with Cayley graph ''Γ''(''G'') equipped with the path metric and a subgroup ''H'' of ''G'', one can construct the coned off Cayley graph $\hat(G,H)$ as follows: For each left coset ''gH'', add a vertex ''v''(''gH'') to the Cayley graph ''Γ''(''G'') and for each element ''x'' of ''gH'', add an edge ''e''(''x'') of length 1/2 from ''x'' to the vertex ''v''(''gH''). This results in a metric space that may not be proper (i.e. closed balls need not be compact).

The definition of a relatively hyperbolic group, as formulated by Bowditch goes as follows. A group ''G'' is said to be hyperbolic relative to a subgroup ''H'' if the coned off Cayley graph $\hat(G,H)$ has the properties:
* It is δ-hyperbolic and
* it is ''fine'': for each integer L, every edge belongs to only finitely many simple cycles of length L.
If only the first condition holds then the group ''G'' is said to be weakly relatively hyperbolic with respect to ''H''.

The definition of the coned off Cayley graph can be generalized to the case of a collection of subgroups and yields the corresponding notion of relative hyperbolicity. A group ''G'' which contains no collection of subgroups with respect to which it is relatively hyperbolic is said to be a non relatively hyperbolic group.

Properties

* If a group ''G'' is relatively hyperbolic with respect to a hyperbolic group ''H'', then ''G'' itself is hyperbolic.

Examples

* Any hyperbolic group, such as a free group of finite rank or the fundamental group of a hyperbolic surface, is hyperbolic relative to the trivial subgroup.
* The fundamental group of a complete hyperbolic manifold of finite volume is hyperbolic relative to its cusp subgroup. A similar result holds for any complete finite volume Riemannian manifold with pinched negative sectional curvature.
* The free abelian group Z² of rank 2 is weakly hyperbolic, but not hyperbolic, relative to the cyclic subgroup Z: even though the graph $\hat(\mathbb^2,\mathbb{Z})$ is hyperbolic, it is not fine.
* The mapping class group of an orientable finite type surface is either hyperbolic (when 3''g''+''n''<5, where ''g'' is the genus and ''n'' is the number of punctures) or is not relatively hyperbolic.
* The automorphism group and the outer automorphism group of a free group of finite rank at least 3 are not relatively hyperbolic.

References

* Mikhail Gromov, ''Hyperbolic groups'', Essays in group theory, Math. Sci. Res. Inst. Publ., 8, 75-263, Springer, New York, 1987.
* Denis Osin
''Relatively hyperbolic groups: Intrinsic geometry, algebraic properties, and algorithmic problems''
arXiv:math/0404040v1 (math.GR), April 2004.
* Benson Farb, ''Relatively hyperbolic groups'', Geom. Funct. Anal. 8 (1998), 810–840.
* Jason Behrstock, Cornelia Druţu, Lee Mosher
''Thick metric spaces, relative hyperbolicity, and quasi-isometric rigidity''
arXiv:math/0512592v5 (math.GT), December 2005.
* Daniel Groves and Jason Fox Manning
''Dehn filling in relatively hyperbolic groups''
arXiv:math/0601311v4 ath.GR January 2007.

Geometric group theory