Summary: An obstruction to the strong relative hyperbolicity of a group
James W. Anderson, Javier Aramayona and Kenneth J. Shackleton
25 December, 2006
We give a simple combinatorial criterion for a group that, when satisfied, implies the group
cannot be strongly relatively hyperbolic. Our criterion applies to several classes of groups, such
as surface mapping class groups, Torelli groups, and automorphism and outer automorphism
groups of free groups.
MSC 20F67 (primary), 20F65 (secondary)
In recent years, the notion of relative hyperbolicity has become a powerful method for establishing
analytic and geometric properties of groups. Relatively hyperbolic groups, first introduced by
Gromov  and then elaborated on by various authors (see Farb , Szczepa´nski , Bowditch
), provide a natural generalization of hyperbolic groups and geometrically finite Kleinian groups.
When a finitely generated group G is strongly hyperbolic relative to a finite collection L1, L2, . . . , Lp
of proper subgroups, it is often possible to deduce that G has a given property provided the sub-
groups Lj have the same property. Examples of such properties include finite asymptotic dimension
(see Osin ), exactness (see Ozawa ), and uniform embeddability in Hilbert space (see Dadar-
lat and Guentner ). In light of this, identifying a strong relatively hyperbolic group structure for
a given group G, or indeed deciding whether or not one can exist, becomes an important objective.