Relatively hyperbolic group

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 $\hat{\Gamma}(G,H)$ 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 $\hat{\Gamma}(G,H)$ 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 $\hat{\Gamma}(\mathbb{Z}^2,\mathbb{Z})$ is hyperbolic, it is not fine.

The mapping class group of an orientable finite type surface is either hyperbolic (when 3g+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 [math.GR], January 2007.

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