# Proof of property of relatively hyperbolic groups on Wikipedia

The Wikipedia page for "Relatively hyperbolic group" lists this as a property of relatively hyperbolic groups:

"If a group $$G$$ is relatively hyperbolic with respect to a hyperbolic group $$H$$, then $$G$$ itself is hyperbolic."

Wikipedia does not provide a proof or reference for this property. Can anybody help me prove it?

I found another equivalent definition for relatively hyperbolic groups in https://arxiv.org/pdf/math/0404040v1.pdf (found in the Wikipedia page's references section). By this equivalent definition $$G$$ is relatively hyperbolic with respect to $$H$$ means that

(1) There exist finite sets $$X,R$$ such that $$X\cup H$$ generates $$G$$ and the kernel of the natural map $$F(X)*H\to G$$ is the normal closure of $$R$$ in $$F:= F(X)*H$$.

(2) For a word $$W \in (X \cup H)^∗$$ such that $$W$$ represents $$1$$ in G, there exists an expression $$W =\prod_{i=1}^{k}f_{i}^{−1}R_{i}f_{i}$$ with the equality in the group $$F$$, where $$R_{i} ∈ R$$ and $$f_i ∈ F$$ for any $$i$$. A function $$f : \mathbb{N} → \mathbb{N}$$ is a relative isoperimetric function with respect to subgroup $$H$$ if for any $$n ∈ \mathbb{N}$$ and any word $$W ∈ (X ∪H)^∗$$ of length $$||W|| ≤ n$$ representing the identity in the group $$G$$, one can write $$W$$ in the form of that expression with $$k ≤ f(n)$$. The second condition is that there exists a linear relative isoperimetric function with this definition.

It seems like it would be easier to prove the property by showing that $$G$$ being relatively hyperbolic with respect to a hyperbolic group $$H$$ implies that $$G$$ has a linear ordinary/non-relative isoperimetric function for $$G$$. $$H$$ being hyperbolic tells us that it is generated by some finite subset $$Y$$. Then $$G$$ being hyperbolic relative to $$H$$ implies that $$X\cup Y$$ is a finite generating set for $$G$$. I wanted to show that there is a finite group presentation for $$G$$ with generating set $$X\cup Y$$ that we can use to get a linear ordinary/non-relative isoperimetric function for $$G$$. I got stuck. Can anyone help me?

We aim to use Lemma 5.3 of that paper to prove that $$G$$ has a linear Dehn function. To do that you first have to choose a suitable (inverse closed) generating set $$X$$ for $$G$$, which involves adding additional generators from the parabolic subgroups $$H$$ to your initial generating set. That ensures that geodesic words over $$X$$ that lie in $$H$$ are written over the generators in $$X \cap H$$.
Now let $$W$$ be a word in the group generators representing the identity in $$G$$. Since are assuming that the parabolic subgroups are hyperbolic, by adjusting $$W$$ using a number of relations that is linear in the length of $$W$$, we may assume that all of the components of $$W$$ that lie in the parabolic subgroups $$H$$ are geodesic words over $$X \cap H$$.
Now, by Lemma 5.3 of the paper, there is a finite subset $$\Phi$$ of non-geodesic words over $$X$$ with certain properties. By adjusting $$W$$ again, using a number of relations linear in the length of $$W$$, we can assume that $$W$$ does not contain any word in $$\Phi$$ as a subword.
At this stage, Lemma 5.3 of the paper tells us that there are constants $$\lambda$$ and $$c$$ such that the derived word $$\widehat{W}$$, which we get by replacing each component in $$H$$ by a single element of $$H$$, is a $$(\lambda,c)$$-quasigeodesic in the derived Cayley graph (which we get by regarding all elements of the parabolic subgroup as generators) without vertex backtracking.
Since $$\widehat{W}$$ represents the identity element, this implies that it has length at most $$c$$ in the derived Cayley graph. But now the bounded coset penetration property (which is a fundamental property of relatively hyperbolic groups, and is stated as Theorem 2.8 of the Antolin Ciobanu paper) implies that there is another constant $$\varepsilon$$ such that $$W$$ has no parabolic components of length greater that $$\varepsilon$$. So $$|W| \le c\epsilon$$ and the result follows.