Relatively hyperbolic group

Last updated June 23, 2023

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.

If a group G is relatively hyperbolic with respect to a group H then it acts as a geometrically finite convergence group on a compact space, its Bowditch boundary

If a group G is relatively hyperbolic with respect to a group H that has solvable word problem, then G has solvable word problem (Farb), and if H has solvable conjugacy problem, then G has solvable conjugacy problem (Bumagin)

If a group G is torsion-free relatively hyperbolic with respect to a group H, and H has a finite classifying space, then so does G (Dahmani)

If a group G is relatively hyperbolic with respect to a group H that satisfies the Farrell-Jones conjecture, then G satisfies the Farrell-jones conjecture (Bartels).

More generally, in many cases (but not all, and not easily or systematically), a property satisfied by all hyperbolic groups and byH can be suspected to be satisfied by G

The isomorphism problem for virtually torsion-free relatively hyperbolic groups when the peripheral subgroups are finitely generated nilpotent (Dahmani, Touikan)

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 free product of a group H with any hyperbolic group, is relatively hyperbolic, relative to H
Limit groups appearing as limits of free groups are relatively hyperbolic, relative to some free abelian subgroups.
The semi-direct product of a free group by an infinite cyclic group is relatively hyperbolic, relative to some canonical subgroups.
Combination theorems and small cancellation techniques allow to construct new examples from previous ones.
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.

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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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