Relatively hyperbolic group explained

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}( Z ^{2,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 with respect to any subgroup.
• 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.