relative to their maximal abelian non-cyclic subgroups. Another proof of this fact
is obtained in [2].
(VI) Let Mod(S) denote the mapping class group corresponding to a surface 5.
Using the isometric action of Mod(S) on the complex of curves introduced in [48],
Masur and Minsky proved that Mod(S) is weakly hyperbolic relative to a finite
collection of stabilizers of certain curves [61]. An alternative proof can be found in
[14]. However, in most cases these groups are not relatively hyperbolic.
(VII) Applying a technique related to small cancellation theory, Kapovich [52]
proved the weak relative hyperbolicity of some Artin groups of extra large type
with respect to certain families of parabolic subgroups. Another result of this
type was obtained by Bahls [4]. He showed that right-angled Coxeter groups are
weakly relatively hyperbolic with respect to natural collections of rank 2 parabolic
(VIII) Finally we mention two combination theorems for weakly relatively hy-
perbolic groups. If G is an HNN-extension of a group H (respectively an amal-
gamated product of H\ and H2) with associated (respectively amalgamated) sub-
groups A and B, then G is weakly hyperbolic relative to H (respectively relative
to {Hi,H2}. Moreover, if H is weakly hyperbolic relative to {A,B} (respectively
Hi is weakly hyperbolic relative to A and H2 is weakly hyperbolic relative to 5) ,
then G is weakly hyperbolic relative to A [69]. In particular, this allows the author
to construct a finitely presented group G which is weakly hyperbolic relative to an
(ordinary) hyperbolic subgroup H and has undecidable word problem.
1.2. Main results
In this section we discuss shortly the main results of our paper. We assume
the reader to be familiar with such notions as Cayley graph, Dehn function, hyper-
bolic group, quasi-geodesic path, etc., and refer to the next chapter for the precise
Let G be a group, X a subset of G, and H an arbitrary subgroup of G. For
simplicity we consider here the case of a single subgroup and refer to Section 2.1
for the general case. We say that X is a relative generating set of G with respect
to i7, if G is generated by the set H U X. In this situation there exists a canonical
e : F - G,
F = H*F(X)
and F(X) is the free group with the basis X. If Ker e is a normal closure of a
subset 1Z C N in F, we say that G has the relative presentation
(1.1) (X,H\R=l,ReK)
The relative presentation (1.1) is finite if the sets X and 1Z are finite. G is said
to be relatively finitely presented with respect to i7, if it admits a finite relative
presentation. Note that G and H need not be finitely presented or even finitely
generated in the usual sense.
Similarly one can define the notion of a relatively finitely generated and a rela-
tively finitely presented group with respect to an arbitrary collection of subgroups.
We begin with the theorem, which shows some restrictions in case G is finitely
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