1.2. MAIN RESULTS 3

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

subgroups.

(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

definitions.

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

homomorphism

e : F - G,

where

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