Proof of property of relatively hyperbolic groups on Wikipedia

Question

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?

Answer 1

I don't think this is particularly easy. It can be proved using the results in this paper of Antolin and Ciobanu.

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.

Answer 2

The statement was proved here, Remark 8.30 (2).