# Small cancellations over relatively hyperbolic groups and embedding theorems

D. V. Osin This work has been partially supported by the RFBR Grants $\sharp$
02-01-00892 and $\sharp$ 03-01-06555.

###### Abstract

In this paper we generalize Olshanskii’s small cancellation theory over
ordinary hyperbolic groups to relatively hyperbolic settings. This theory is
then used to prove various embedding theorem for countable groups. For
instance, we show that any countable torsion free group can be embedded into a
finitely generated group with exactly 2 conjugacy classes. In particular, this
gives the affirmative answer to the well–known question of the existence of a
finitely generated group $G$ other than $\mathbb{Z}/2\mathbb{Z}$ such that all
nontrivial elements of $G$ are conjugate.

##  1 Introduction

Originally the notion of relative hyperbolicity was proposed by Gromov [11] in
order to generalize various examples of algebraic and geometric nature such as
Kleinian groups, fundamental groups of hyperbolic manifolds of pinched
negative curvature, small cancellation quotients of free products, etc. It has
been extensively studied in the last several years from different points of
view. The main aim of this paper is to generalize the small cancellation
theory over hyperbolic groups developed by Olshanskii [22] to relatively
hyperbolic settings. Our approach is based on author’s papers [23, 24, 25],
where the necessary background is provided. In the present paper we apply
small cancellations over relatively hyperbolic groups to prove embedding
theorems for countable groups. Further applications of our methods can be
found in [1, 2, 4, 19, 20, 26].

In the paper [13], Higman, B.H. Neumann, and H. Neumann proved that any
countable group $G$ can be embedded into a countable group $B$ such that every
two elements of the same order are conjugate in $B$. We notice that the group
$B$ in [13] is constructed as a union of infinite number of subsequent
HNN–extensions and thus $B$ is never finitely generated. On the other hand,
any countable group can be embedded into a $2$–generated group [13]. Our first
theorem is a natural generalization of both of these results. For a group $G$,
we denote by $\pi(G)$ the set of all finite orders of elements of $G$.

######  Theorem 1.1.

Any countable group $G$ can be embedded into a $2$–generated group $C$ such
that any two elements of the same order are conjugate in $C$ and
$\pi(G)=\pi(C)$.

######  Corollary 1.2.

Any countable torsion–free group can be embedded into a (torsion–free)
$2$–generated group with exactly $2$ conjugacy classes.

Since the number of finitely generated subgroups in any 2–generated groups is
at most countable and the number of all torsion–free finitely generated groups
is uncountable, we have

######  Corollary 1.3.

There exists an uncountable set of pairwise non–isomorphic torsion–free
$2$–generated groups with exactly $2$ conjugacy classes.

We note that the question of the existence of an (at least one) finitely
generated group with exactly $2$ conjugacy classes other than
$\mathbb{Z}/2\mathbb{Z}$ was open until now. It can be found, for example, in
[16, Problem 9.10] or in [3, Problem FP20]. (Positive solution has been
announced by Ivanov in 1989 [14, 15] but the complete proof has never been
published.) Corollary 1.3 provides first examples of such groups. Starting
with the group $G=\mathbb{Z}/p^{n-2}\mathbb{Z}\times H$ for $n\geq 3$, where
$p$ is a prime number and $H$ is a torsion–free group, we can generalize the
previous result.

######  Corollary 1.4.

For any $n\in\mathbb{N}$, $n\geq 2$, there is an uncountable set of pairwise
non–isomorphic finitely generated groups with exactly $n$ conjugacy classes.

For large enough prime numbers $n$, the first examples of finitely generated
infinite periodic groups with exactly $n$ conjugacy classes were constructed
by Ivanov (see [21]) as limits of hyperbolic groups (although hyperbolicity
was not used explicitly). Here we say that $G$ is a limit of hyperbolic groups
if there exists a finitely generated free group $F$ and a series of normal
subgroups $N_{1}\lhd N_{2}\lhd\ldots$ of $F$ such that $G\cong F/N$ for
$N=\bigcup\limits_{i=1}^{\infty}N_{i}$ and each of the groups $F/N_{i}$,
$i=1,2,\ldots$ is hyperbolic. In contrast it is impossible to construct a
finitely generated group other than $\mathbb{Z}/2\mathbb{Z}$ with exactly $2$
conjugacy classes in this way.

Indeed suppose that a finitely generated group $G$ has exactly $2$ conjugacy
classes. If $G$ is not torsion–free, then $G$ is a group of exponent $p$ for
some prime $p$ as the orders of all nontrivial elements of $G$ are equal. If
$p=2$, $G$ is abelian and hence is isomorphic to $\mathbb{Z}/2\mathbb{Z}$. In
case $p>2$, there exist non–trivial elements $g,t\in G$ such that

$t^{-1}gt=g^{2}.$ (1)

The equality $g^{2^{p}-1}=t^{-p}gt^{p}g^{-1}=gg^{-1}=1$ implies $2^{p}-1\equiv
0(mod\;p)$. However, by the Fermat Little Theorem, we have $2^{p}-2\equiv
0(mod\;p)$, which contradicts the previous equality. Assume now that $G$ is
torsion–free. If $G$ is a limit of hyperbolic groups $F/N_{i}$,
$i=1,2,\ldots$, then for some $i$ large enough, there are elements $g,t\in
F/N_{i}$ of infinite order that satisfy (1). This leads to a contradiction
again since the equality of type (1) is impossible in a hyperbolic group if
the order of $g$ is infinite [10, 11].

Another theorem from [13] states that any countable group $G$ can be embedded
into a countable divisible group $D$. We recall that a group $D$ is said to be
divisible if for every element $d\in D$ and every positive integer $n$, the
equation $x^{n}=d$ has a solution in $D$. A natural example of a divisible
groups is $\mathbb{Q}$. The question of the existence of a finitely generated
divisible group was open during a long time. The first examples of such a type
were constructed by Guba [12] (see also [21]).

Later Mikhajlovskii and Olshanskii [18] constructed a more general example of
a finitely generated verbally complete group, that is a group $W$ such that
for every nontrivial freely reduced word $w(x_{i})$ in the alphabet
$x_{1}^{\pm 1},x_{2}^{\pm 1},\ldots$ and every $v\in W$, the equation
$w(x_{i})=v$ has a solution in $W$. That is, there are elements
$v_{1},v_{2},\ldots\in W$ such that $w(v_{i})=v$ in $W$, where $w(v_{i})$ is
the word obtained from $w(x_{i})$ by substituting $v_{i}$ for $x_{i}$,
$i=1,2\ldots$. Comparing these results one may ask whether any countable group
can be embedded into a finitely generated divisible (or verbally complete)
group. The next theorem provides the affirmative answer.

######  Theorem 1.5.

Any countable group $H$ can be embedded into a $2$–generated verbally complete
group $W$. Moreover, if $H$ is torsion–free, then $W$ can be chosen to be
torsion–free.

Note that the condition $\pi(G)=\pi(W)$ can not be ensured in Theorem 1.5.
Indeed, it is easy to show that if a divisible group $W$ contains a nontrivial
element of finite order, then $\pi(W)=\mathbb{N}$. As above, we obtain

######  Corollary 1.6.

There exists an uncountable set of pairwise non–isomorphic $2$–generated
verbally complete groups.

Acknowledgment. The author is grateful to A. Minasyan and the anonymous
referee for careful reading of the manuscript and useful remarks.

##  2 Outline of the method

In this section we give the proofs of Theorem 1.5 and Theorem 1.1 modulo
technical results which are obtained in Sections 4-6. We assume the reader to
be familiar with the notion of a relatively hyperbolic group and refer to the
next section for precise definitions.

Let $G$ be a group that is hyperbolic relative to a collection of subgroups
$\\{H_{\lambda}\\}_{\lambda\in\Lambda}$. We divide the set of all elements of
$G$ into two subsets as follows. An element $g\in G$ is said to be parabolic
if $g$ is conjugate to an element of $H_{\lambda}$ for some
$\lambda\in\Lambda$. Otherwise $g$ is said to be hyperbolic. Recall also that
a group is elementary  if it contains a cyclic subgroup of finite index. The
following result concerning maximal elementary subgroups is proved in
[24][Theorem 4.3, Corollary 1.7].

######  Theorem 2.1.

Let $G$ be a group hyperbolic relative to a collection of subgroups
$\\{H_{\lambda}\\}_{\lambda\in\Lambda}$, $g$ a hyperbolic element of infinite
order of $G$. Then the following conditions hold.

  1. 1.

The element $g$ is contained in a unique maximal elementary subgroup
$E_{G}(g)$ of $G$, where

$E_{G}(g)=\\{f\in G\;:\;f^{-1}g^{n}f=g^{\pm n}\;{\rm
for\;some\;}n\in\mathbb{N}\\}.$

  2. 2.

The group $G$ is hyperbolic relative to the collection
$\\{H_{\lambda}\\}_{\lambda\in\Lambda}\cup\\{E_{G}(g)\\}$.

Given a subgroup $H\leq G$, we denote by $H^{0}$ the set of all hyperbolic
elements of infinite order in $H$. Recall also that two elements $f,g\in
G^{0}$ are said to be commensurable (in G) if $f^{k}$ is conjugate to $g^{l}$
in $G$ for some non–zero $k,l$.

######  Definition 2.2.

A subgroup $H\leq G$ is called suitable, if there exist two non–commensurable
elements $f_{1},f_{2}\in H^{0}$ such that $E_{G}(f_{1})\cap E_{G}(f_{2})=1$.

The next lemma is proved in Section 6.

######  Lemma 2.3.

Let $G$ be a group hyperbolic relative to a collection of subgroups
$\\{H_{\lambda}\\}_{\lambda\in\Lambda}$, $H$ a suitable subgroup of $G$. Then
there exist infinitely many pairwise non–commensurable (in G) elements
$h_{1},h_{2},\ldots\in H^{0}$ such that for all $i=1,2,\ldots$,
$E_{G}(h_{i})=\langle h_{i}\rangle$. In particular, $E_{G}(h_{i})\cap
E_{G}(h_{j})=\\{1\\}$ whenever $i\neq j$.

Our main tool is the following theorem proved in Section 6. The proof is based
on a certain small cancellation techniques developed in Sections 4 and 5.

######  Theorem 2.4.

Let $G$ be a group hyperbolic relative to a collection of subgroups
$\\{H_{\lambda}\\}_{\lambda\in\Lambda}$, $H$ a suitable subgroup of $G$, and
$t_{1},\ldots,t_{m}$ arbitrary elements of $G$. Then there exists an
epimorphism $\eta\colon G\to\overline{G}$ such that:

  1. 1.

The group $\overline{G}$ is hyperbolic relative to
$\\{\eta(H_{\lambda})\\}_{\lambda\in\Lambda}$.

  2. 2.

For any $i=1,\ldots,m$, we have $\eta(t_{i})\in\eta(H)$.

  3. 3.

The restriction of $\eta$ to $\bigcup\limits_{\lambda\in\Lambda}H_{\lambda}$
is injective.

  4. 4.

$\eta(H)$ is a suitable subgroup of $\overline{G}$.

  5. 5.

An element of $\overline{G}$ has finite order, only if it is an image of an
element of finite order in $G$. In particular, if all hyperbolic elements of
$G$ have infinite order, then all hyperbolic elements of $\overline{G}$ have
infinite order.

The next theorem is proved in [13][Corollary 1.4]. For finitely generated
groups this result was also proved by Dahmani in [7]. It is worth noting that
we use the theorem for infinitely generated groups in this paper.

######  Theorem 2.5.

Suppose that a group $G$ is hyperbolic relative to a collection of subgroups
$\\{H_{\lambda}\\}_{\lambda\in\Lambda}\cup\\{K\\}$ and for some
$\nu\in\Lambda$, there exists a monomorphism $\iota\colon K\to H_{\nu}$. Then
the HNN–extension

$\widetilde{G}=\langle G,t\;|\;t^{-1}kt=\iota(k),\;k\in K\rangle$ (2)

is hyperbolic relative to $\\{H_{\lambda}\\}_{\lambda\in\Lambda}$.

Theorems 1.5 and 1.1 can be obtained in a uniform way from the following
result.

######  Theorem 2.6.

Suppose that $R$ is a countable group such that for any elementary group $E$
satisfying the condition $\pi(E)\subseteq\pi(R)$, there exists a subgroup of
$R$ isomorphic to $E$. Then there is an embedding of $R$ into a 2–generated
group $S=S(R)$ such that any element of $S$ is conjugate to an element of $R$
in $S$. In particular, $\pi(S)=\pi(R)$.

###### Proof.

The desired group $S$ is constructed as an inductive limit of relatively
hyperbolic groups as follows. Let us set

$G(0)=R\ast F(x,y),$

where $F(x,y)$ is the free group of rank $2$ generated by $x$ and $y$. We
enumerate all elements of

$R=\\{1=r_{0},r_{1},r_{2},\ldots\\}$

and

$G(0)=\\{1=g_{0},g_{1},g_{2},\ldots\\}.$

Suppose that for some $i\geq 0$, the group $G(i)$ has already been constructed
together with an epimorphism $\xi_{i}\colon G(0)\to G(i)$. We use the same
notation for elements $x,y,r_{0},r_{1},\ldots,g_{0},g_{1},\ldots$ and their
images under $\xi_{i}$ in $G(i)$. Assume that $G(i)$ satisfies the following
conditions. (It is straightforward to check these conditions for $G(0)$ and
the identity map $\xi_{0}\colon G(0)\to G(0)$.)

  1. (i)

The restriction of $\xi_{i}$ to the subgroup $R$ is injective. In what follows
we identify $R$ with its image in $G(i)$.

  2. (ii)

$G(i)$ is hyperbolic relative to $R$.

  3. (iii)

The elements $x$ and $y$ generate a suitable subgroup of $G(i)$.

  4. (iv)

All hyperbolic elements of $G(i)$ have infinite order. In particular,
$\pi(G(i))=\pi(R)$.

  5. (v)

The elements $g_{0},\ldots,g_{i}$ are parabolic in $G(i)$.

  6. (vi)

In the group $G(i)$, the elements $r_{0},\ldots,r_{i}$ are contained in the
subgroup generated by $x$ and $y$.

The group $G(i+1)$ is obtained from $G(i)$ in two steps.

Step 1. Let us take the element $g_{i+1}$ and construct a group $G(i+1/2)$ as
follows. If $g_{i+1}$ is a parabolic element of $G(i)$, we set
$G(i+1/2)=G(i)$. If $g_{i+1}$ is hyperbolic, the order of $g_{i+1}$ is
infinite by (iv). Furthermore, since
$\pi(E_{G(i)}(g_{i+1}))\subseteq\pi(G(i))=\pi(R)$, there is a monomorphism
$\iota\colon E_{G(i)}(g_{i+1})\to R$. Then we take the HNN–extension

$G(i+1/2)=\langle G(0),t\;|\;t^{-1}et=\iota(e),\,e\in
E_{G(i)}(g_{i+1})\rangle.$

In both cases $G(i+1/2)$ is hyperbolic relative to $R$. Indeed this is obvious
in the first case and follows from the second assertion of Theorem 2.1 and
Theorem 2.5 in the second one. Note also that all hyperbolic elements of
$G(i+1/2)$ have infinite order. (In the second case this immediately follows
from the description of periodic elements in HNN–extensions [17, Ch. IV,
Theorem 2.4].)

Step 2. First we wish to show that the subgroup generated by $x$ and $y$ is
suitable in $G(i+1/2)$. This is obvious in case $g_{i+1}$ is parabolic in
$G(i)$, so we consider the second case only. Since $\langle x,y\rangle$ is
suitable in $G(i)$ by (iii), Lemma 2.3 yields the existence of infinitely many
pairwise non–commensurable (in $G(i)$) hyperbolic elements $h_{j}\in\langle
x,y\rangle$ of infinite order, $j=1,2\ldots$, such that
$E_{G(i)}(h_{j})=\langle h_{j}\rangle$. At most one of these elements is
commensurable with $g_{i+1}$ in $G(i)$. Therefore, there exist two
non–commensurable in $G(i)$ hyperbolic elements of infinite order, say
$h_{1},h_{2}\in\langle x,y\rangle$, such that $h_{j}$ is not commensurable
with $g_{i+1}$ in $G(i)$ for $j=1,2$. In particular, $h_{j}$, $j=1,2$, is not
conjugate to an element of $E_{G(i)}(g_{i+1})$ as $\langle g_{i+1}\rangle$ has
finite index in $E_{G(i)}(g_{i+1})$. According to Britton’s Lemma on
HNN–extensions [17, Ch. 5, Sec.2], this implies that $h_{1}$ and $h_{2}$ are
hyperbolic and non–commensurable in $G(i+1/2)$. Furthermore, if for some
$j=1,2$, $n\in\mathbb{N}$, and $u\in G(i+1/2)$, we have
$u^{-1}h_{j}^{n}u=h_{j}^{\pm n}$, then $u\in G(i)$ by the Britton Lemma. Thus
the explicit description of maximal elementary subgroups from the first
assertion of Theorem 2.1 yields the equality
$E_{G(i+1/2)}(h_{j})=E_{G(i)}(h_{j})$ for $j=1,2$. Finally since
$E_{G(i)}(h_{j})=\langle h_{j}\rangle$ and $h_{1}$, $h_{2}$ are
non–commensurable, we have

$E_{G(i+1/2)}(h_{1})\cap E_{G(i+1/2)}(h_{2})=E_{G(i)}(h_{1})\cap
E_{G(i)}(h_{2})=\langle h_{1}\rangle\cap\langle h_{2}\rangle=\\{1\\}.$

By Definition 2.2 this means that the subgroup generated by $x$ and $y$ is
suitable in $G(i+1/2)$.

We now apply Theorem 2.4 to the group $G=G(i+1/2)$, the subgroup $H=\langle
x,y\rangle\leq G(i+1/2)$, and the set of elements $\\{t,r_{i+1}\\}$. Let
$G(i+1)=\overline{G}$, where $\overline{G}$ is the quotient group provided by
Theorem 2.4. Since $t$ becomes an element of $\langle x,y\rangle$ in $G(i+1)$,
there is a naturally defined epimomorphism $\xi_{i+1}\colon G(0)\to G(i+1)$.
Using Theorem 2.4 it is straightforward to check properties (i)–(vi) for
$G(i+1)$. This completes the inductive step.

Let $N_{i}$ denote the kernel of $\xi_{i}$. Observe that $N_{1},N_{2},\ldots$
form an increasing normal series and set $S=G(0)/N$, where
$N=\bigcup_{i=1}^{\infty}N_{i}$. By (i) the subgroup $R$ is embedded into $S$.
Further it is easy to see that $S$ is $2$–generated. Indeed, $G(0)$ is
generated by $x,y,r_{1},r_{2},\ldots$. Condition (vi) yields $r_{i}\in\langle
x,y\rangle$ in $S$ for any $i\in\mathbb{N}$. Thus $S$ is generated by $x$ and
$y$. Finally let $s$ be an element of $S$. We take an arbitrary preimage $g\in
G(0)$ of $s$. Then the image of the element $g$ becomes parabolic at a certain
step according to (v). Thus $s$ is conjugate to an element of $R$ in $S$. The
theorem is proved. ∎

It remains to derive Theorems 1.5 and 1.1.

###### Proof of Theorem 1.1..

Let $\mathcal{E}$ denote the free product of all elementary groups $E$ (taken
up to isomorphism) such that $\pi(E)\subseteq\pi(G)$. We set
$G^{\ast}=G\ast\mathcal{E}$. By a theorem from [13], we can embed $G^{\ast}$
into an (infinitely generated) group $R$ such that all elements of the same
order are conjugate in $R$ and

$\pi(R)=\pi(G^{\ast})=\pi(G).$ (3)

We now apply Theorem 2.6 and embed the group $R$ into a 2-generated group
$C=S(R)$ such that any element of $C$ is conjugate to an element of $R$. As
all elements of the same order are conjugate in $R$, this is so in $C$. The
equality $\pi(C)=\pi(G)$ follows from (3) as $\pi(C)=\pi(R)$ by Theorem 2.6. ∎

###### Proof of Theorem 1.5..

First note that any countable group $G$ can be embedded into an infinitely
generated countable verbally complete group $R$ in the following way. (The
idea comes from the proof of the Higman–Neumann–Neumann theorem on embeddings
into divisible groups.) We denote by $F=F(a_{1},a_{2},\ldots)$ the free group
with basis $a_{1},a_{2},\ldots$. Let us enumerate the set of all pairs

$\\{p_{1},p_{2},\ldots\\}=\\{(v,g)\;:\;v\in F\setminus\\{1\\},\,g\in
G\setminus\\{1\\}\\}.$

Starting with the group $G$ we first set $G^{\ast}=G$ if $G$ is torsion–free,
and $G^{\ast}=G\ast E_{1}\ast E_{2}\ast\ldots$, where
$\\{E_{1},E_{2},\ldots\\}$ is the set of all elementary groups (up to
isomorphism), otherwise. Further we construct a sequence of groups
$G=U_{0}\leq U_{1}\leq\ldots$ as follows. Suppose that for some $i\geq 0$, the
group $U_{i}$ has already been constructed and take $p_{i+1}=(v,g)$. There are
two possibilities to consider.

1) The element $g$ has infinite order. Then we define $U_{i+1}$ to be the free
product of $U_{i}$ and $F$ with the amalgamated subgroups $\langle g\rangle$
and $\langle v\rangle$.

2) The order of $g$ is $n<\infty$. It is well–known [17, Theorem 5.2, Ch. 4]
that the order of the element $v$ in the group $H=\langle
a_{1},a_{2},\ldots\;|\;v^{n}=1\rangle$ equals $n$. Thus the free product of
$U_{i}$ and $H$ with amalgamated subgroups $\langle g\rangle$ and $\langle
v\rangle$ is well–defined. We set $U_{i+1}=U_{i}\ast_{\langle g\rangle=\langle
v\rangle}H$.

Now let $U(G)=\bigcup\limits_{i=0}^{\infty}U_{i}$. Obviously $G$ embeds in
$U(G)$, $U(G)$ is countable and torsion–free whenever $G$ is torsion–free, and
any equation of type $w(x_{i})=g$, where $w(x_{i})$ is a word in the alphabet
$x_{1}^{\pm 1},x_{2}^{\pm 1},\ldots$ and $g\in G$, has a solution in $U(G)$.
Finally we consider the sequence of groups $R_{1}\leq R_{2}\leq\ldots$, where
$R_{1}=U(G)$ and $R_{i+1}=U(R_{i})$, $i=1,2\ldots$. Clearly the group
$R=\bigcup\limits_{i=0}^{\infty}R_{i}$ is countable, verbally complete,
torsion–free whenever $G$ is torsion–free, and contains a copy of every
elementary group $E$ such that $\pi(E)\subseteq\pi(G)$. Let $W=S(R)$ be the
group provided by Theorem 2.6.

Consider an equations $w(x_{i})=v$ for some $v\in W$. By Theorem 2.6, there is
an element $t\in W$ such that $t^{-1}vt\in R$. Since $R$ is verbally complete,
there is a solution $x_{1}=r_{1}$, $x_{2}=r_{2}$, $\ldots$ to the equation
$w(x_{i})=t^{-1}vt$ in $R$. Clearly $x_{1}=tr_{1}t^{-1}$,
$x_{2}=tr_{2}t^{-1}$, $\ldots$ is a solution to the equation $w(x_{i})=v$. ∎

##  3 Preliminaries

Some conventions and notation. We write $W\equiv V$ to express the
letter–for–letter equality of words $W$ and $V$ in some alphabet. If a word
$W$ decomposes as $W\equiv V_{1}UV_{2}$, we call $V_{1}$ (respectively,
$V_{2}$) a prefix (respectively, suffix) of $W$. For elements $g$, $t$ of a
group $G$, $g^{t}$ denotes the element $t^{-1}gt$. Recall that a subset $X$ of
a group $G$ is said to be symmetric if for any $x\in X$, we have $x^{-1}\in
X$. In this paper all generating sets of groups under consideration are
supposed to be symmetric.

Word metrics and Cayley graphs. Let $G$ be a group generated by a (symmetric)
set $\mathcal{A}$. Recall that the Cayley graph $\Gamma(G,\mathcal{A})$ of a
group $G$ with respect to the set of generators $\mathcal{A}$ is an oriented
labelled 1–complex with the vertex set $V(\Gamma(G,\mathcal{A}))=G$ and the
edge set $E(\Gamma(G,\mathcal{A}))=G\times\mathcal{A}$. An edge $e=(g,a)$ goes
from the vertex $g$ to the vertex $ga$ and has label ${\rm\bf Lab\,}(e)\equiv
a$. As usual, we denote the origin and the terminus of the edge $e$, i.e., the
vertices $g$ and $ga$, by $e_{-}$ and $e_{+}$ respectively. Given a
combinatorial path $p=e_{1}e_{2}\ldots e_{k}$ in the Cayley graph
$\Gamma(G,\mathcal{A})$, where $e_{1},e_{2},\ldots,e_{k}\in
E(\Gamma(G,\mathcal{A}))$, we denote by ${\rm\bf Lab\,}(p)$ its label. By
definition, ${\rm\bf Lab\,}(p)\equiv{\rm\bf Lab\,}(e_{1}){\rm\bf
Lab\,}(e_{2})\ldots{\rm\bf Lab\,}(e_{k}).$ We also denote by
$p_{-}=(e_{1})_{-}$ and $p_{+}=(e_{k})_{+}$ the origin and the terminus of $p$
respectively. The length $l(p)$ of $p$ is the number of edges of $p$.

Associated to $\mathcal{A}$ is the so–called word metric on $G$. More
precisely, the length $|g|_{\mathcal{A}}$ of an element $g\in G$ is defined to
be the length of a shortest word in $\mathcal{A}$ representing $g$ in $G$. By
abuse of notation, we also write $|W|_{\mathcal{A}}$ to denote the lengths of
the element of $G$ represented by a word $W$ in the alphabet $\mathcal{A}$.
This is to be distinguished from the lengths of the word $W$ itself, which is
denoted by $\|W\|$.

The word metric on $G$ is defined by
$dist_{\mathcal{A}}(f,g)=|f^{-1}g|_{\mathcal{A}}$. We also denote by
$dist_{\mathcal{A}}$ the natural extension of the word metric to the Cayley
graph $\Gamma(G,\mathcal{A})$.

Van Kampen Diagrams. Recall that a van Kampen diagram $\Delta$ over a
presentation

$G=\langle\mathcal{A}\;|\;\mathcal{O}\rangle$ (4)

is a finite oriented connected 2–complex endowed with a labelling function
${\rm\bf Lab\,}:E(\Delta)\to\mathcal{A}$, where $E(\Delta)$ denotes the set of
oriented edges of $\Delta$, such that ${\rm\bf Lab\,}(e^{-1})\equiv({\rm\bf
Lab\,}(e))^{-1}$. Labels and lengths of paths are defined as in the case of
Cayley graphs. Given a cell $\Pi$ of $\Delta$, we denote by $\partial\Pi$ the
boundary of $\Pi$; similarly, $\partial\Delta$ denotes the boundary of
$\Delta$. The labels of $\partial\Pi$ and $\partial\Delta$ are defined up to a
cyclic permutation. An additional requirement is that for any cell $\Pi$ of
$\Delta$, the boundary label ${\rm\bf Lab\,}(\partial\Pi)$ is equal to a
cyclic permutation of a word $P^{\pm 1}$, where $P\in\mathcal{O}$. Sometimes
it is convenient to use the notion of the so–called $0$–refinement in order to
assume diagrams to be homeomorphic to a disc. We do not explain here this
notion and refer the interested reader to [21, Ch. 4].

The van Kampen Lemma states that a word $W$ over the alphabet $\mathcal{A}$
represents the identity in the group given by (4) if and only if there exists
a simply–connected planar diagram $\Delta$ over (4) such that ${\rm\bf
Lab\,}(\partial\Delta)\equiv W$ [17, Ch. 5, Theorem 1.1].

For every van Kampen diagram $\Delta$ over (4) and ani fixed vertex $o$ of
$\Delta$, there is a (unique) combinatorial map $\gamma\colon
Sk^{(1)}(\Delta)\to\Gamma(G,\mathcal{A})$ that preserves labels and
orientation of edges and maps $o$ to the vertex $1$ of
$\Gamma(G,\mathcal{A})$.

Hyperbolic spaces. Here we briefly discuss some properties of hyperbolic
spaces used in this paper. For more details we refer to [6, 10, 11].

One says that a metric space $M$ is $\delta$–hyperbolic for some $\delta\geq
0$ (or simply hyperbolic) if for any geodesic triangle $T$ in $M$, any side of
$T$ belongs to the union of the closed $\delta$–neighborhoods of the other two
sides.

Recall that a path $p$ in a metric space is called
$(\lambda,c)$–quasi–geodesic for some $\lambda>0$, $c\geq 0$, if

$dist(q_{-},q_{+})\geq\lambda l(q)-c$

for any subpath $q$ of $p$. Let $p$ be a path in a van Kampen diagram $\Delta$
over (4). We need the following result about quasi–geodesics in hyperbolic
spaces (see, for example, [6, Ch. III. H, Theorem 1.7]).

######  Lemma 3.1.

For any $\delta\geq 0$, $\lambda>0$, $c\geq 0$, there exists a constant
$\kappa=\kappa(\delta,\lambda,c)$ with the following property. If $M$ is a
$\delta$–hyperbolic space and $p,q$ are $(\lambda,c)$–quasi–geodesic paths in
$M$ with same endpoints, then $p$ and $q$ belong to the closed
$\kappa$–neighborhoods of each other.

The next result can be easily derived from the definition of a hyperbolic
space by drawing the diagonal.

######  Lemma 3.2.

Let $M$ be a $\delta$–hyperbolic metric space, $Q$ a geodesic quadrangle in
$M$. Then each side of $Q$ belongs to the closed $2\delta$–neighborhood of the
other three sides.

From Lemma 3.1 and Lemma 3.2, we immediately obtain

######  Corollary 3.3.

For any $\delta\geq 0$, $\lambda>0$, $c\geq 0$, there exists a constant
$K=K(\delta,\lambda,c)$ with the following property. Let $Q$ be a quadrangle
in a $\delta$–hyperbolic space whose sides are $(\lambda,c)$–quasi–geodesic.
Then each side of $Q$ belongs to the closed $K$–neighborhood of the union of
the other three sides.

Indeed it suffices to set $K=2(\kappa+\delta)$, where $\kappa$ is provided by
Lemma 3.1.

The proof of the next lemma is also straightforward (see [22, Lemma 1.7]).

######  Lemma 3.4.

Let $Q$ be a geodesic quadrangle with sides $a,b,c,d$ in a $\delta$-hyperbolic
space. Suppose that $l(a)\geq 4\max\\{l(b),\,l(d)\\}$. Then there exist
subsegments $p$ , $q$ of the sides $a$ and $b$, respectively, such that
$\min\\{l(p),\,l(q)\\}\geq\frac{7}{20}l(a)-8\delta$ and ${\rm
dist\,}(p_{\pm},q_{\pm})\leq 8\delta$.

Passing from geodesic to quasi-geodesic quadrangles one can easily obtain the
following. The proof is straightforward and consists of replacing the quasi-
geometric quadrangle with a geodesic one having the same vertices, application
of Lemma 3.4, and then Lemma 3.1. We leave details to the reader.

######  Corollary 3.5.

Let $Q$ be a $(\lambda,c)$–quasi-geodesic quadrangle with sides $a,b,c,d$ in a
$\delta$-hyperbolic space, $\kappa=\kappa(\lambda,c)$ the constant provided by
Lemma 3.1. Suppose that $l(a)\geq(4\max\\{l(b),\,l(d)\\}+c)/\lambda$. Then
there exist subsegments $p$, $q$ of the sides $a$ and $b$, respectively, such
that $\min\\{l(p),\,l(q)\\}\geq\frac{7}{20}(\lambda l(a)-c)-8\delta-2\kappa$
and ${\rm dist\,}(p_{\pm},q_{\pm})\leq 8\delta+2\kappa$.

The following lemma is also well known (see, for example, [6, CH. III.H,
Theorem 1.13]). Recall that a path in a metric space is said to be $k$–local
geodesic if any its subpath of length at most $k$ is geodesic.

######  Lemma 3.6.

Let $r$ be a $k$–local geodesic in a $\delta$–hyperbolic metric space for some
$k>8\delta$. Then $r$ is $(1/3,2\delta)$–quasi–geodesic.

Relatively hyperbolic groups. There are many equivalent definitions of
relatively hyperbolic groups (see [5, 8, 23] and references therein). In this
paper we use the isoperimetric characterization suggested in [23].

More precisely, let $G$ be a group, $\\{H_{\lambda}\\}_{\lambda\in\Lambda}$ a
collection of subgroups of $G$, $X$ a subset of $G$. We say that $X$ is a
relative generating set of $G$ with respect to
$\\{H_{\lambda}\\}_{\lambda\in\Lambda}$ if $G$ is generated by $X$ together
with the union of all $H_{\lambda}$. (In what follows we $X$ to be symmetric.)
In this situation the group $G$ can be regarded as a quotient group of the
free product

$F=\left(\ast_{\lambda\in\Lambda}H_{\lambda}\right)\ast F(X),$ (5)

where $F(X)$ is the free group with the basis $X$. Let $N$ denote the kernel
of the natural homomorphism $F\to G$. If $N$ is a normal closure of a subset
$\mathcal{Q}\subseteq N$ in the group $F$, we say that $G$ has relative
presentation

$\langle X,\;H_{\lambda},\lambda\in\Lambda\;|\;\mathcal{Q}\rangle.$ (6)

If $\sharp\,X<\infty$ and $\sharp\,\mathcal{Q}<\infty$, the relative
presentation (6) is said to be finite and the group $G$ is said to be finitely
presented relative to the collection of subgroups
$\\{H_{\lambda}\\}_{\lambda\in\Lambda}$.

Set

$\mathcal{H}=\bigsqcup\limits_{\lambda\in\Lambda}(H_{\lambda}\setminus\\{1\\}).$
(7)

Given a word $W$ in the alphabet $X\cup\mathcal{H}$ such that $W$ represents
$1$ in $G$, there exists an expression

$W=_{F}\prod\limits_{i=1}^{k}f_{i}^{-1}Q_{i}^{\pm 1}f_{i}$ (8)

with the equality in the group $F$, where $Q_{i}\in\mathcal{Q}$ and $f_{i}\in
F$ for $i=1,\ldots,k$. The smallest possible number $k$ in a representation of
the form (8) is called the relative area of $W$ and is denoted by
$Area^{rel}(W)$.

######  Definition 3.7.

A group $G$ is hyperbolic relative to a collection of subgroups
$\\{H_{\lambda}\\}_{\lambda\in\Lambda}$ if $G$ is finitely presented relative
to $\\{H_{\lambda}\\}_{\lambda\in\Lambda}$ and there is a constant $L>0$ such
that for any word $W$ in $X\cup\mathcal{H}$ representing the identity in $G$,
we have $Area^{rel}(W)\leq L\|W\|$.

In particular, $G$ is an ordinary hyperbolic group if $G$ is hyperbolic
relative to the trivial subgroup. An equivalent definition says that $G$ is
hyperbolic if it is generated by a finite set $X$ and the Cayley graph
$\Gamma(G,X)$ is hyperbolic. In the relative case these approaches are not
equivalent, but we still have the following [23, Theorem 1.7].

######  Lemma 3.8.

Suppose that $G$ is a group hyperbolic relative to a collection of subgroups
$\\{H_{\lambda}\\}_{\lambda\in\Lambda}$. Let $X$ be a finite relative
generating set of $G$ with respect to $\\{H_{\lambda}\\}_{\lambda\in\Lambda}$.
Then the Cayley graph $\Gamma(G,X\cup\mathcal{H})$ of $G$ with respect to the
generating set $X\cup\mathcal{H}$ is a hyperbolic metric space.

Observe also that the relative area of a word $W$ representing $1$ in $G$ can
be defined geometrically via van Kampen diagrams. Let $G$ be a group given by
the relative presentation (6) with respect to a collection of subgroups
$\\{H_{\lambda}\\}_{\lambda\in\Lambda}$. We denote by $\mathcal{S}$ the set of
all words in the alphabet $\mathcal{H}$ representing the identity in the
groups $F$ defined by (5). Then $G$ has the ordinary (non–relative)
presentation

$G=\langle X\cup\mathcal{H}\;|\;\mathcal{S}\cup\mathcal{Q}\rangle.$ (9)

A cell in van Kampen diagram $\Delta$ over (9) is called a $\mathcal{Q}$–cell
if its boundary is labelled by a word from $\mathcal{Q}$. We denote by
$N_{\mathcal{Q}}(\Delta)$ the number of $\mathcal{Q}$–cells of $\Delta$.
Obviously given a word $W$ in $X\cup\mathcal{H}$ that represents $1$ in $G$,
we have

$Area^{rel}(W)=\min\limits_{{\rm\bf Lab\,}(\partial\Delta)\equiv
W}\\{N_{\mathcal{Q}}(\Delta)\\},$

where the minimum is taken over all van Kampen diagrams with boundary label
$W$.

$H_{\lambda}$–components. Finally we are going to recall an auxiliary
terminology introduced in [23], which plays an important role in our paper.
Let $G$ be a group, $\\{H_{\lambda}\\}_{\lambda\in\Lambda}$ a collection of
subgroups of $G$, $X$ a finite generating set of $G$ with respect to
$\\{H_{\lambda}\\}_{\lambda\in\Lambda}$, $q$ a path in the Cayley graph
$\Gamma(G,X\cup\mathcal{H})$. A subpath $p$ of $q$ is called an
$H_{\lambda}$–component for some $\lambda\in\Lambda$ (or simply a component)
of $q$, if the label of $p$ is a word in the alphabet
$H_{\lambda}\setminus\\{1\\}$ and $p$ is not contained in a bigger subpath of
$q$ with this property. Two $H_{\lambda}$–components $p_{1},p_{2}$ of a path
$q$ in $\Gamma(G,X\cup\mathcal{H})$ are called connected if there exists a
path $c$ in $\Gamma(G,X\cup\mathcal{H})$ that connects some vertex of $p_{1}$
to some vertex of $p_{2}$ and ${{\rm\bf Lab\,}(c)}$ is a word consisting of
letters from $H_{\lambda}\setminus\\{1\\}$. In algebraic terms this means that
all vertices of $p_{1}$ and $p_{2}$ belong to the same coset $gH_{\lambda}$
for a certain $g\in G$. Note that we can always assume $c$ to have length at
most $1$, as every nontrivial element of $H_{\lambda}$ is included in the set
of generators. An $H_{\lambda}$–component $p$ of a path $q$ is called isolated
if no distinct $H_{\lambda}$–component of $q$ is connected to $p$.

The lemma below is a simplification of Lemma 2.27 from [23].

######  Lemma 3.9.

Suppose that $G$ is a group that is hyperbolic relative to a collection of
subgroups $\\{H_{\lambda}\\}_{\lambda\in\Lambda}$, $X$ a finite generating set
of $G$ with respect to $\\{H_{\lambda}\\}_{\lambda\in\Lambda}$. Then there
exists a constant $K>0$ and a finite subset $\Omega\subseteq G$ such that the
following condition holds. Let $q$ be a cycle in $\Gamma(G,X\cup\mathcal{H})$,
$p_{1},\ldots,p_{k}$ a set of isolated $H_{\lambda}$–components of $q$ for
some $\lambda\in\Lambda$, $g_{1},\ldots,g_{k}$ the elements of $G$ represented
by the labels of $p_{1},\ldots,p_{k}$ respectively. Then for any
$i=1,\ldots,k$, $g_{i}$ belongs to the subgroup $\langle\Omega\rangle\leq G$
and the lengths of $g_{i}$ with respect to $\Omega$ satisfy the inequality

$\sum\limits_{i=1}^{k}|g_{i}|_{\Omega}\leq Kl(q).$

##  4 Small cancellation conditions

The main aim of this and the following four sections is to generalize the
small cancellation theory over hyperbolic groups developed by Olshanskii in
[22] to relatively hyperbolic groups. The fact that the Cayley graph
$\Gamma(G,X)$ of a hyperbolic group $G$ generated by a finite set $X$ is a
hyperbolic metric space plays the key role in [22]. Lemma 3.8 allows to extend
this theory to the case of relatively hyperbolic groups. However this
extension is not straightforward as the Cayley graph
$\Gamma(G,X\cup\mathcal{H})$ defined in the previous section is not necessary
locally finite.

Roughly speaking, one can divide results about small cancellation conditions
from [22] into three classes. The first class consists of results about
diagrams over presentations satisfying small cancellation conditions, which do
not use local finiteness of Cayley graphs at all. They can be stated and
proved in our settings without any essential changes. The main result of this
part is Lemma 4.3 stated below.

Proof of results from the second class do not use local finiteness of the
Cayley graph either, but they do employ certain facts concerning geometric and
algebraic properties of ordinary hyperbolic groups. These results can also be
reproved with minor changes modulo the paper [23], where the corresponding
facts about relatively hyperbolic groups are proved. For convenience of the
reader we provide self-contained proofs in Sections 5 and 6.

Finally results of the third type explain how to chose words of some specific
form satisfying small cancellation conditions. Unlike in ordinary small
cancellation theory over a free group, verifying small cancellation conditions
over hyperbolic groups is much harder and local finiteness of Cayley graphs is
essentially used in [22]. Our approach here is different and is explained in
Section 7.

Given a set of words $\mathcal{R}$ in an alphabet $\mathcal{A}$, we say that
$\mathcal{R}$ is symmetrized, if for any $R\in\mathcal{R}$, $\mathcal{R}$
contains all cyclic shifts of $R^{\pm 1}$. Further let $G$ be a group
generated by a set $\mathcal{A}$. We say that a word $R$ is
$(\lambda,c)$–quasi–geodesic in $G$, if any path in the Cayley graph
$\Gamma(G,\mathcal{A})$ labelled $R$ is $(\lambda,c)$–quasi–geodesic.

######  Definition 4.1.

Let $G$ be a group generated by a set $\mathcal{A}$, $\mathcal{R}$ a
symmetrized set of words in $\mathcal{A}$. For $\varepsilon>0$, a subword $U$
of a word $R\in\mathcal{R}$ is called an $\varepsilon$–piece if there exists a
word $R^{\prime}\in\mathcal{R}$ such that:

  1. (1)

$R\equiv UV$, $R^{\prime}\equiv U^{\prime}V^{\prime}$, for some
$V,U^{\prime},V^{\prime}$;

  2. (2)

$U^{\prime}=YUZ$ in $G$ for some words $Y,Z$ in $\mathcal{A}$ such that
$\max\\{\|Y\|,\,\|Z\|\\}\leq\varepsilon$;

  3. (3)

$YRY^{-1}\neq R^{\prime}$ in the group $G$.

Similarly, a subword $U$ of $R\in\mathcal{R}$ is called an
$\varepsilon^{\prime}$–piece if:

  1. ($1^{\prime}$)

$R\equiv UVU^{\prime}V^{\prime}$ for some $V,U^{\prime},V^{\prime}$;

  2. ($2^{\prime}$)

$U^{\prime}=YU^{\pm 1}Z$ in the group $G$ for some $Y,Z$ satisfying
$\max\\{\|Y\|,\|Z\|\\}\leq\varepsilon$.

Recall that a word $R$ in $\mathcal{A}$ is said to be $(\lambda,c)$-quasi-
geodesic, if some (equivalently, any) path labelled by $R$ in
$\Gamma(G,\mathcal{A})$ is $(\lambda,c)$-quasi-geodesic.

######  Definition 4.2.

We say that the set $\mathcal{R}$ satisfies the
$C(\varepsilon,\mu,\lambda,c,\rho)$–condition for some $\varepsilon\geq 0$,
$\mu>0$, $\lambda>0$, $c\geq 0$, $\rho>0$, if

  1. (1)

$\|R\|\geq\rho$ for any $R\in\mathcal{R}$;

  2. (2)

any word $R\in\mathcal{R}$ is $(\lambda,c)$–quasi–geodesic;

  3. (3)

for any $\varepsilon$–piece of any word $R\in\mathcal{R}$, the inequality
$\max\\{\|U\|,\,\|U^{\prime}\|\\}<\mu\|R\|$ holds (using the notation of
Definition 4.1).

Further the set $\mathcal{R}$ satisfies the
$C_{1}(\varepsilon,\mu,\lambda,c,\rho)$–condition if in addition the condition
$(3)$ holds for any $\varepsilon^{\prime}$–piece of any word
$R\in\mathcal{R}$.

$u$$w$$\Pi$$q_{1}$$q_{2}$$s_{1}$$s_{2}$$\Gamma$$\Xi$ Figure 1: Contiguity
subdiagram of a cell to the boundary of the diagram.

Suppose that $G$ is a group defined by (4). Given a set of words
$\mathcal{R}$, we consider the quotient group of $G$ represented by

$G_{1}=\langle\mathcal{A}\;|\;\mathcal{O}\cup\mathcal{R}\rangle.$ (10)

A cell in a van Kampen diagram over (10) is called an $\mathcal{R}$–cell if
its boundary label is a word from $\mathcal{R}$.

Let $\Delta$ be a van Kampen diagram over (10), $\Pi$ an $\mathcal{R}$–cell of
$\Delta$. Suppose that there is a simple closed paths

$p=s_{1}q_{1}s_{2}q_{2}$ (11)

in $\Delta$, where $q_{1}$ (respectively $q_{2}$) is a subpath of
$\partial\Pi$ (respectively $\partial\Delta$) and

$\max\\{\|s_{1}\|,\,\|s_{2}\|\\}\leq\varepsilon$ (12)

for some constant $\varepsilon$. By $\Gamma$ we denote the subdiagram of
$\Delta$ bounded by $p$ (see Fig. 1). If $\Gamma$ contains no
$\mathcal{R}$–cells, we say that $\Gamma$ is an $\varepsilon$–contiguity
subdiagram of $\Pi$ to $\partial\Delta$, and $q_{1}$ is the contiguity arc of
$\Pi$ to $\partial\Delta$. The ratio $l(q_{1})/l(\partial\Pi)$ is called the
contiguity degree of $\Pi$ to $\partial\Delta$ (respectively $\Pi_{1}$ to
$\Pi_{2}$) and is denoted by $(\Pi,\Gamma,\partial\Delta)$.

Recall that a van Kampen diagram $\Delta$ over (10) is said to be reduced if
$\Delta$ has minimal number of $\mathcal{R}$–cells among all diagrams over
(10) having the same boundary label. A path $p$ in $\Delta$ is called
$(\lambda,c)$-quasi-geodesic, if its label is $(\lambda,c)$-quasi-geodesic,
i.e., some (equivalently, any) path in $\Gamma(G,\mathcal{A})$ with the same
label is $(\lambda,c)$-quasi-geodesic.

The following is an analogue of the well known Greendlinger Lemma.

######  Lemma 4.3.

Let $G$ be a group with a presentation
$G=\langle\mathcal{A}\,|\,\mathcal{O}\rangle$. Suppose that the Cayley graph
$\Gamma(G,\mathcal{A})$ of $G$ is hyperbolic. Then for any $\lambda\in(0,1]$,
there exists $\mu>0$ such that for every $c\geq 0$, there are $\varepsilon\geq
0$ and $\rho>0$ with the following property. Let $\mathcal{R}$ be a
symmetrized set of words in $\mathcal{A}$ satisfying the
$C(\varepsilon,\mu,\lambda,c,\rho)$–condition, $\Delta$ a reduced van Kampen
diagram over the presentation (10) whose boundary can be represented as
$\partial\Delta=q^{1}\ldots q^{r}$ for some $1\leq r\leq 4$, where
$q^{1},\ldots,q^{r}$ are $(\lambda,c)$–quasi–geodesic. Assume that $\Delta$
has at least one $\mathcal{R}$–cell. Then there exists an $\mathcal{R}$–cell
$\Pi$ in $\Delta$ and $\varepsilon$–contiguity subdiagrams
$\Gamma_{1},\ldots,\Gamma_{r}$ of $\Pi$ to $q^{1},\ldots,q^{r}$, respectively,
such that

$\sum\limits_{i=1}^{r}(\Pi,\Gamma_{i},q^{i})>1-23\mu.$

This lemma is proved in [22] (see Lemma 6.6 there) under the assumption that
the group $G$ is hyperbolic, i.e., $\Gamma(G,\mathcal{A})$ is hyperbolic and
$\mathcal{A}$ is finite. However the finiteness of $\mathcal{A}$ is not used
in the proof, and can be removed from the assumptions of the lemma.

More precisely, to prove Lemma 4.3 one needs to repeat Section 5 of and
Section 6 (from Lemma 6.1 to Lemma 6.6) of [22] verbatim. We note that all
results of Section 5 and Lemma 6.1 from Section 6 of [22] are stated and
proved for arbitrary (not necessarily finite) presentations of arbitrary (not
necessarily hyperbolic) groups. The only place where hyperbolicity of $G$ is
used in Sections 5 and 6 of [22] is the proof of Lemma 6.2. The assumption
that $G$ is hyperbolic is used there to conclude that the Cayley graph
$\Gamma(G,\mathcal{A})$ is hyperbolic, which allows to apply Lemmas 1.8 and
1.9 from [22]. However Lemmas 1.8 and 1.9 are stated for arbitrary (not
necessarily locally finite) hyperbolic metric spaces. Thus, assuming
hyperbolicity of $\Gamma(G,\mathcal{A})$, the proof of Lemma 6.2 and the
proofs of subsequent Lemmas 6.3–6.6 in [22] work without any changes.

Lemma 4.3 in the form stated above plays an important role in [26], where it
is used to analyze centralizers of elements in the quotient group. However in
this paper we only need the following particular case corresponding to $r=1$.

######  Corollary 4.4.

Suppose that the Cayley graph $\Gamma(G,\mathcal{A})$ of a group
$G=\langle\mathcal{A}\,|\,\mathcal{O}\rangle$ is hyperbolic. Then for any
$\lambda\in(0,1]$, there exist $\mu>0$ such that for every $c\geq 0$, there
are $\varepsilon\geq 0$ and $\rho>0$ with the following property. Let
$\mathcal{R}$ be a symmetrized set of words in $\mathcal{A}$ satisfying the
$C(\varepsilon,\mu,\lambda,c,\rho)$–condition, $\Delta$ a reduced van Kampen
diagram over the presentation (10) whose boundary is
$(\lambda,c)$–quasi–geodesic. Assume that $\Delta$ has at least one
$\mathcal{R}$–cell. Then there exists an $\mathcal{R}$–cell $\Pi$ in $\Delta$
and $\varepsilon$–contiguity subdiagram $\Gamma$ such that
$(\Pi,\Gamma,\partial\Delta)>1-23\mu.$

##  5 Hyperbolicity of the quotient

Our next goal is to show that adding relations satisfying sufficiently strong
small cancellation conditions preserves relative hyperbolicity. Throughout
this section let $G$ be a group hyperbolic relative to a collection of
subgroups $\\{H_{\lambda}\\}_{\lambda\in\Lambda}$, $X$ a finite relative
generating set of $G$ with respect to $\\{H_{\lambda}\\}_{\lambda\in\Lambda}$.
We set $\mathcal{A}=X\cup\mathcal{H}$ and
$\mathcal{O}=\mathcal{S}\cup\mathcal{Q}$, where $\mathcal{S}$ and
$\mathcal{Q}$ are defined as in (9). In the proof of the lemma below we follow
the idea from [22, Lemma 6.7] with little changes.

######  Lemma 5.1.

For any $\lambda\in(0,1]$, $c\geq 0$, $N>0$, there exist $\mu>0$,
$\varepsilon\geq 0$, and $\rho>0$ such that for any finite symmetrized set of
words $\mathcal{R}$ satisfying the
$C(\varepsilon,\mu,\lambda,c,\rho)$–condition, the following hold.

  1. 1.

The group $G_{1}$ defined by (10) is hyperbolic relative to the collection of
images of subgroups $H_{\lambda},\lambda\in\Lambda$, under the natural
homomorphism $G\to G_{1}$. In particular, the Cayley graph of $G_{1}$ with
respect to the generating set $\mathcal{A}$ is hyperbolic.

  2. 2.

The restriction of the natural homomorphism $\gamma\colon G\to G_{1}$ to the
subset of elements of length at most $N$ with respect to the generating set
$\mathcal{A}$ is injective.

###### Proof.

Let us choose the constants $\mu$, $\varepsilon$, $\rho$ according to
Corollary 4.4.

For a word $W$ in the alphabet $X\cup\mathcal{H}$ that represents $1$ in
$G_{1}$, we denote by $Area_{1}^{rel}(W)$ its relative area, that is the
minimal number $k$ in a representation of type

$W=_{F}\prod\limits_{i=1}^{k}f_{i}R_{i}^{\pm 1}f_{i}^{-1}$

with the equality in the group $F$ defined by (5), where
$R_{i}\in\mathcal{R}\cup\mathcal{Q}$. Similarly, by $Area^{rel}(W)$ we denote
the relative area of a word $W$ representing $1$ in $G$, i.e., the minimal
number $k$ in the above decomposition, where $R_{i}\in\mathcal{R}$.

As $G$ is hyperbolic relative to $\\{H_{\lambda}\\}_{\lambda\in\Lambda}$,
there exists a constant $L>0$ such that for any word $W$ in $X\cup\mathcal{H}$
representing $1$ in $G$, we have $Area^{rel}(W)\leq L\|W\|$. To prove the
lemma it suffices to show that there is a constant $\alpha>0$ such that for
any word $W$ representing $1$ in $G_{1}$, $Area_{1}^{rel}(W)\leq\alpha\|W\|$.
We are going to prove this inequality for

$\alpha=\frac{3L}{\lambda-47\mu}.$

We proceed by induction on the length of $W$. Let $p$ be a path labelled $W$
in $\Gamma(G,X\cup\mathcal{H})$. Suppose first that $p$ is not
$(1/2,0)$–quasi–geodesic. Passing to a cyclic shift of $W$ if necessary, we
may assume that $p=p_{0}p_{1}$, where $p_{0}$ is a subpath of $p$ such that
$dist_{X\cup\mathcal{H}}((p_{0})_{-},(p_{0})_{+})<l(p_{0})/2$. Let $q$ be a
geodesic path in $\Gamma(G,X\cup\mathcal{H})$ that goes from $(p_{0})_{-}$ to
$(p_{0})_{+}$. Thus $l(q)<l(p_{0})/2$. We denote by $U_{0}$, $U_{1}$, and $V$
the labels of $p_{0}$, $p_{1}$, and $q$ respectively. Then

$W=_{F}(U_{0}V^{-1})(VU_{1}),$

where $U_{0}V^{-1}$ represents $1$ in $G$ and $VU_{1}$ represents $1$ in
$G_{1}$. Obviously we have

$\|VU_{1}\|\leq l(p)-l(p_{0})+l(q)<\|W\|-\frac{l(p_{0})}{2}$

and

$\|U_{0}V^{-1}\|\leq l(p_{0})+l(q)<\frac{3l(p_{0})}{2}.$

Using the inductive hypothesis, we obtain

$\begin{array}[]{rl}Area_{1}^{rel}(W)&\leq
Area_{1}^{rel}(VU_{1})+Area^{rel}(U_{0}V^{-1})\\\ &\\\
&<\alpha\left(\|W\|-\frac{1}{2}l(p_{0})\right)+\frac{3}{2}Ll(p_{0})<\alpha\|W\|\end{array}$

as $\alpha>3L$.

Now suppose that $p$ is a $(1/2,0)$–quasi–geodesic path in
$\Gamma(G,X\cup\mathcal{H})$. Since
$C(\varepsilon,\mu,\lambda,c,\rho)$–condition implies
$C(\varepsilon,\mu,1/2,c,\rho)$ whenever $\lambda>1/2$, it suffices to prove
the lemma for $\lambda\leq 1/2$. So we may assume that $p$ is
$(\lambda,c)$–quasi–geodesic as well. Let $\Delta$ be a reduced diagram over
(10) such that $\partial\Delta$ is labelled $W$. Assume that $\Delta$ has at
least one $\mathcal{R}$–cell. (Otherwise the lemma is obviously true.) By
Corollary 4.4 there exists a $\mathcal{R}$–cell $\Pi$ and a contiguity
subdiagram $\Gamma$ of $\Pi$ to $\partial\Delta$ with
$(\Pi,\Gamma,\partial\Delta)>1-23\mu$. Let
$\partial\Gamma=s_{1}q_{1}s_{2}q_{2}$, where $q_{1}$ (respectively $q_{2}$) is
a subpath of $\partial\Pi$ (respectively $\partial\Delta$) and
$\max\\{\|s_{1}\|,\,\|s_{2}\|\\}\leq\varepsilon$ (see Fig. 1). Let also
$\partial\Pi=q_{1}u$ and $\partial\Delta=wq_{2}$.

Note that perimeter of the subdiagram $\Xi$ of $\Delta$ bounded by the path
$s_{2}^{-1}us_{1}^{-1}w$ is smaller than perimeter of $\Delta$ if $\rho$ is
large enough and $\mu$ is close to zero. Indeed as $\Gamma$ contains no
$\mathcal{R}$–cells, we can regard $s_{1}q_{1}s_{2}q_{2}$ as a cycle in the
Cayley graph $\Gamma(G,X\cup\mathcal{H})$. Thus,

$\begin{array}[]{rl}l(q_{2})&\geq
dist_{X\cup\mathcal{H}}((q_{2})_{-},(q_{2})_{+})\geq
dist_{X\cup\mathcal{H}}((q_{1})_{-},(q_{1})_{+})-2\varepsilon\geq\lambda
l(q_{1})-c-2\varepsilon\\\ &\\\
&\geq\lambda(1-23\mu)l(\partial\Pi)-c-2\varepsilon.\end{array}$ (13)

On the other hand,

$l(s_{2}^{-1}us_{1}^{-1})\leq 2\varepsilon+23\mu l(\partial\Pi).$ (14)

Since $l(\partial\Pi)\geq\rho$, if $\rho$ is big enough and $\mu$ is close to
zero, the right side of (13) is greater than the right side of (14) and hence
$l(\partial\Xi)<l(\partial\Delta)$.

Therefore, by induction the total number $n_{1}$ of $\mathcal{R}$– and
$\mathcal{Q}$–cells in $\Xi$ is at most

$\begin{array}[]{rl}n_{1}&\leq\alpha
l(\partial\Xi)\leq\alpha\big{(}\|W\|-l(q_{2})+l(s_{2}^{-1}us_{1}^{-1})\big{)}\\\
&\\\
&\leq\alpha\big{(}\|W\|-l(\partial\Pi)(\lambda-46\mu)+c+4\varepsilon\big{)}.\end{array}$
(15)

Furthermore, as $q_{2}$ is $(1/2,0)$–quasi–geodesic in
$\Gamma(G,X\cup\mathcal{H})$, we have

$l(q_{2})\leq 2dist_{X\cup\mathcal{H}}((q_{2})_{-},(q_{2})_{+})\leq
2(dist_{X\cup\mathcal{H}}((q_{1})_{-},(q_{1})_{+})+2\varepsilon)\leq
2l(\partial\Pi)+4\varepsilon.$

Therefore the perimeter of $\Gamma$ satisfies

$l(\partial\Gamma)\leq 2\varepsilon+l(q_{1})+l(q_{2})\leq
3l(\partial\Pi)+6\varepsilon.$

Hence we may assume that the number $n_{2}$ of $\mathcal{Q}$–cells of $\Gamma$
is at most

$n_{2}\leq Ll(\partial\Gamma)\leq
3L(l(\partial\Pi)+2\varepsilon)\leq\alpha(l(\partial\Pi)(\lambda-47\mu)+2\varepsilon).$
(16)

Finally, combining (15) and (16), we obtain

$Area_{1}^{rel}(W)\leq n_{1}+n_{2}\leq\alpha(\|W\|-\mu
l(\partial\Pi)+c+6\varepsilon)\leq\alpha\|W\|$

whenever $l(\partial\Pi)$ is big enough. This completes the proof of relative
hyperbolicity of $G_{1}$. The hyperbolicity of the Cayley graph follows from
Lemma 3.8.

Note that if $\mu<1/50$, the inequality (13) implies

$\|W\|\geq\frac{1}{2}\lambda\rho-c-2\varepsilon$

for every non–trivial word $W$ which is geodesic in $G$ and represents $1$ in
$G_{1}$. Therefore, the second statement of the lemma holds if we assume
$\rho\geq 2(N+c+2\varepsilon)/\lambda$. ∎

##  6 Torsion in the quotient

Our next goal is to describe periodic elements in the quotient group (10) of a
relatively hyperbolic group $G$. To this end we need an auxiliary result.
Recall that for an element $g$ of a group $G$ generated by a set
$\mathcal{A}$, the translation number of $g$ with respect to $\mathcal{A}$ is
defined to be

$\tau_{\mathcal{A}}(g)=\lim\limits_{n\to\infty}\frac{|g^{n}|_{\mathcal{A}}}{n}.$

This limit always exists and is equal to
$\inf\limits_{n}(|g|_{\mathcal{A}}/n)$ [9]. The lemma below can be found in
[23, Theorem 4.43].

######  Lemma 6.1.

There exists $d>0$ such that for any hyperbolic element of infinite order
$g\in G$ we have $\tau_{X\cup\mathcal{H}}(g)\geq d$.

For our goals, even a stronger result is necessary.

######  Lemma 6.2.

There exist $1\geq\alpha>0$, $a\geq 0$ with the following property. Suppose
that $g$ is a hyperbolic element of $G$ of infinite order such that $g$ has
the smallest length among all elements of the conjugacy class $g^{G}$. Denote
by $U$ a shortest word in the alphabet $X\cup\mathcal{H}$ representing $g$.
Then for any $n\in\mathbb{N}$, any path in $\Gamma(G,X\cup\mathcal{H})$
labelled $U^{n}$ is $(\alpha,a)$–quasi–geodesic.

###### Proof.

Recall that $\Gamma(G,X\cup\mathcal{H})$ is hyperbolic by Lemma 3.8. First
suppose that $|g|_{X\cup\mathcal{H}}=\|U\|>8\delta$, where $\delta$ is the
hyperbolicity constant of $\Gamma(G,X\cup\mathcal{H})$. Since $g$ is a
shortest element in $g^{G}$ and $U$ is a shortest word representing $g$, the
path $p$ labelled $U^{n}$ is $k$–local geodesic in
$\Gamma(G,X\cup\mathcal{H})$, $k>8\delta$, for any $n$. Therefore, by Lemma
3.6, $p$ is $(1/3,2\delta)$–quasi–geodesic.

Further if $|g|_{X\cup\mathcal{H}}=\|U\|\leq 8\delta$, then for any
$n\in\mathbb{N}$, we have

$|g^{n}|_{X\cup\mathcal{H}}\geq
n\inf\limits_{i}\left(\frac{1}{i}|g^{i}|_{X\cup\mathcal{H}}\right)\geq
nd\geq\frac{d}{8\delta}n|g|_{X\cup\mathcal{H}},$

where $d$ is the constant provided by Lemma 6.1. Hence the path $p$ labelled
$U^{n}$ is $\left(\frac{d}{8\delta},8\delta\right)$–quasi–geodesic. It remains
to set $\alpha=\min\\{\frac{1}{3},\,\frac{d}{8\delta}\\}$ and $a=8\delta$. ∎

######  Lemma 6.3.

For any $\lambda\in(0,1]$, $c\geq 0$ there are $\mu>0$, $\varepsilon\geq 0$,
and $\rho>0$ such that the following condition holds. Suppose that
$\mathcal{R}$ is a symmetrized set of words in $\mathcal{A}$ satisfying the
$C_{1}(\varepsilon,\mu,\lambda,c,\rho)$–condition. Then every element of
finite order in the group $G_{1}$ given by (10) is the image of an element of
finite order of $G$.

###### Proof.

Let us fix $\lambda,c>0$. Observe that the
$C_{1}(\varepsilon,\mu,\lambda,c,\rho)$–condition becomes stronger as
$\lambda$ increases and $c$ decreases. Hence it suffices to prove the lemma
assuming that $\lambda<\alpha$ and $c>a$ for $\alpha$ and $a$ provided by
Lemma 6.2. Let us choose constants $1>\mu>0$, $\varepsilon>0$, $\rho>0$ such
that

  1. ($\ast$)

the conclusion of Corollary 4.4 holds.

Again we note that the $C_{1}(\varepsilon,\mu,\lambda,c,\rho)$–condition
becomes stronger as $\mu$ decreases and $\varepsilon,\rho$ increase. Hence we
may decrease $\mu$ and increase $\rho$ and $\varepsilon$ if necessary without
violating ($\ast$). In particular, without loss of generality, we may assume
that

$\varepsilon>2\kappa+8\delta,$ (17)

where $\kappa=\kappa(\lambda,c)$ is the constant provided by Lemma 3.1 and
$\delta$ is the hyperbolicity constant of $\Gamma(G,X\cup\mathcal{H})$. We fix
$\varepsilon$ from now on.

Suppose that $g$ is an element of order $n>0$ in $G_{1}$ such that its
preimage has infinite order in $G$. Assume also that $g$ has the smallest
length among all elements from the conjugacy class $g^{G_{1}}$. Denote by $U$
a shortest word in the alphabet $X\cup\mathcal{H}$ representing $g$ in
$G_{1}$. Then there exists a diagram $\Delta$ over (10) with boundary label
$U^{n}$. By Lemma 6.2, the label of $\partial\Delta$ is $(\lambda,c)$–quasi-
geodesic (in $G$) and $\Delta$ contains at least one $\mathcal{R}$-cell. (For
otherwise $\Delta$ is a diagram over (4) and $U^{n}=1$ in $G$.) Hence by
Corollary 4.4 there exists an $\mathcal{R}$–cell $\Pi$ in $\Delta$ and
$\varepsilon$–contiguity subdiagram $\Gamma$ of $\Pi$ to $\partial\Delta$ such
that $(\Pi,\Gamma,\partial\Delta)>1-23\mu.$

Let $\partial\Gamma=s_{1}q_{1}s_{2}q_{2}$ as on Fig. 1. Since $\Gamma$ has no
$\mathcal{R}$-cells, we may think of $s_{1}q_{1}s_{2}q_{2}$ as a qadrangle in
$\Gamma(G,X\cup\mathcal{H})$, where $q_{i}$ is $(\lambda,c)$-quasi-geodesic,
and $l(s_{i})\leq\varepsilon$ for $i=1,2$. Therefore we have

$\begin{array}[]{rl}l(q_{2})&\geq
dist_{X\cup\mathcal{H}}((q_{2})_{-},(q_{2})_{+})\geq
dist_{X\cup\mathcal{H}}((q_{1})_{-},(q_{1})_{+})-l(s_{1})-l(s_{2})\\\ &\\\
&\geq\lambda
l(q_{2})-c-2\varepsilon\geq\lambda(1-23\mu)l(\partial\Pi)-c-2\varepsilon.\end{array}$
(18)

In particular, choosing $\rho$ large enough and $\mu$ small enough we can make
$l(q_{2})$ as large as we want. (Recall that $l(\partial\Pi)\geq\rho$ by the
$C_{1}(\varepsilon,\mu,\lambda,c,\rho)$–condition.)

Passing from $U$ to a cyclic shift of $U^{\pm}1$ if necessary, we may assume
that ${\rm\bf Lab\,}(q_{2})$ is a prefix of $U^{n}$ . We now have three cases
to consider. Our goal is to show that neither of them is possible whenever
$\rho$ is large enough and $\mu$ is small enough.

Case 1. Suppose that $l(q_{2})\geq 4\|U\|/3$. This allows us to find two long
disjoint but equal subwords of ${\rm\bf Lab\,}(q_{2})$. More precisely, we can
decompose ${\rm\bf Lab\,}(q_{2})$ as ${\rm\bf Lab\,}(q_{2})\equiv
WV_{1}WV_{2}$, where $\lambda^{2}l(q_{2})/5\leq\|W\|\leq\lambda^{2}l(q_{2})/4$
and $\|V_{1}\|\geq l(q_{2})/3$. Let $q_{2}=w_{1}v_{1}w_{2}v_{2}$ be the
corresponding decomposition of the path $q_{2}$. Corollary 3.3 applied to the
quadrangle $s_{1}q_{1}s_{2}q_{2}$ implies that there is a point $o\in q_{1}$
such that $dist_{X\cup\mathcal{H}}(o,(w_{1})_{+})\leq K+\varepsilon$, where
$K$ depends on $\lambda$, $c$, and $\delta$ only. Let $q_{1}^{-1}=r_{1}t$,
where $(r_{1})_{+}=o$. Thus we have

$dist_{X\cup\mathcal{H}}((r_{1})_{\pm},(w_{1})_{\pm})\leq K+\varepsilon$ (19)

Similarly one can find a subpath $r_{2}^{\xi}$, $\xi=\pm 1$, of $q_{1}^{-1}$
such that

$dist_{X\cup\mathcal{H}}((r_{2}^{\xi})_{\pm},(w_{2})_{\pm})\leq
K+\varepsilon.$ (20)

We note that $r_{1}$ and $r_{2}$ are disjoint. Indeed otherwise $r_{1}$ passes
through $(r_{2})_{-}$ or $(r_{2})_{+}$. For definiteness, assume that
$(r_{2})_{-}\in r_{1}$. Then we have

$\begin{array}[]{rl}l(r_{1})&\leq\frac{1}{\lambda}(dist_{X\cup\mathcal{H}}((r_{1})_{-},(r_{1})_{+})+c)\leq\frac{1}{\lambda}(l(w_{1})+2\varepsilon+K+c)\\\
&\\\ &=\frac{1}{\lambda}(\|W\|+2\varepsilon+K+c)\leq\lambda
l(q_{2})/4+(2\varepsilon+K+c)/\lambda.\end{array}$

On the other hand,

$\begin{array}[]{rl}l(r_{1})&\geq
dist_{X\cup\mathcal{H}}((r_{1})_{-},(r_{2})_{-})\geq\lambda
l(w_{1}v_{1})-c-2\varepsilon-K\geq\\\ &\\\
&\lambda\|V_{1}\|-c-2\varepsilon-K\geq\lambda
l(q_{2})/3-c-2\varepsilon-K.\end{array}$

These inequalities contradict each other if $l(q_{2})$ is large enough. As
explained before, the later condition can always be ensured by choosing
sufficiently small $\mu$ and sufficiently large $\rho$ (see (18)).

Thus we have a decomposition $q_{1}^{-1}=r_{1}t_{1}r_{2}^{\xi}t_{2}$, $\xi=\pm
1$. Let ${\rm\bf Lab\,}(q_{1})^{-1}\equiv R_{1}T_{1}R_{2}T_{2}$ be the
corresponding decomposition of the label. By (19) and (20), we have
$R_{1}=Y_{1}WZ_{1}$ and $R_{2}=Y_{2}W^{\pm 1}Z_{2}$ in $G$, where
$\|Y_{i}\|,\|X_{i}\|\leq K+\varepsilon$ for $i=1,2$. Hence $R_{1}=YR_{2}^{\pm
1}Z$ in $G$, where $\|Y\|,\|Z\|\leq 2(K+\varepsilon)$. Without loss of
generality, we may assume that words $Y$ and $Z$ are geodesic in $G$. Let
$a_{1}ya_{2}z$ be the corresponding rectangle in $\Gamma(G,X\cup\mathcal{H})$,
where ${\rm\bf Lab\,}(a_{1})\equiv R_{1}^{-1}$, ${\rm\bf Lab\,}(y)\equiv Y$,
${\rm\bf Lab\,}(a_{2})\equiv R_{2}^{\pm 1}$, ${\rm\bf Lab\,}(z)\equiv Z$.
Recall that $R_{1},R_{2}$ are subwords of the $(\lambda,c)$–quasi-geodesic
word ${\rm\bf Lab\,}(\partial\Pi)$. Hence $a_{1},a_{2}$ are
$(\lambda,c)$–quasi-geodesic. Moreover, we have

$\begin{array}[]{rl}l(a_{1})&=\|R_{1}\|\geq
dist_{X\cup\mathcal{H}}((r_{1})_{-},(r_{1})_{-})\\\ &\\\ &\geq
dist_{X\cup\mathcal{H}}((w_{1})_{-},(w_{1})_{+})-l(s_{1})-dist_{X\cup\mathcal{H}}((r_{1})_{+},(w_{1})_{+})\\\
&\\\ &\geq\lambda l(w_{1})-c-2\varepsilon-K=\lambda\|W\|-c-2\varepsilon-K\\\
&\\\
&\geq\lambda^{3}l(q_{2})/5-c-2\varepsilon-K\geq\lambda^{3}(1-23\mu)l(\partial\Pi)/5-c-2\varepsilon-K.\end{array}$

In particular, if $\mu$ is small enough and $\rho$ is large enough, we have
$l(a)>(4\max\\{l(y),l(z)\\}+c)/\lambda$. This allows us to apply Corollary
3.5. Thus we obtain subsegments $a^{\prime},b^{\prime}$ of the sides $a$ and
$b$, respectively, such that the distances between the corresponding ends of
these subsegments is at most $8\delta+2\kappa$. Let $A={\rm\bf
Lab\,}(a^{\prime})$, $B={\rm\bf Lab\,}(b^{\prime})$. Then $A=CB^{\pm 1}D$ in
$G$, where $\|C\|,\|D\|\leq 8\delta+2\kappa<\varepsilon$ by (17) and

$\begin{array}[]{rl}\min\\{\|A\|,\|B\|\\}&\geq\frac{7}{20}(\lambda
l(a_{1})-c)-8\delta-2\kappa\\\ &\\\
&\geq\frac{7}{20}(\lambda(\lambda^{3}(1-23\mu)l(\partial\Pi)/5-c-2\varepsilon-K)-c)-8\delta-2\kappa\geq\mu
l(\partial\Pi)\end{array}$

if $\mu$ is small enough and $\rho$ is large enough. Since $A$ and $B$ are
disjoint subwords of ${\rm\bf Lab\,}(\partial\Pi)$, we obtain a contradiction
with the $C_{1}(\varepsilon,\mu,\lambda,c,\rho)$–condition.

Case 2. Suppose that $\|U\|\leq l(q_{2})\leq 4\|U\|/3$, i.e. ${\rm\bf
Lab\,}(q_{2})\equiv UV$ for some (may be empty) word $V$, $\|V\|\leq\|U\|/3$.
Note that ${\rm\bf Lab\,}(q_{2})={\rm\bf Lab\,}(s_{2}^{-1}us_{1}^{-1})$ in
$G_{1}$ (see Fig. 1), hence $g={\rm\bf Lab\,}(s_{2}^{-1}us_{1}^{-1})V^{-1}$ in
$G_{1}$. Since $U$ is the shortest word representing $g$ in $G_{1}$, we obtain

$\|U\|\leq 2\varepsilon+23\mu l(\partial\Pi)+\|V\|\leq 2\varepsilon+23\mu
l(\partial\Pi)+\|U\|/3.$

Consequently,

$\|U\|\leq 3(2\varepsilon+23\mu l(\partial\Pi))/2.$ (21)

On the other hand, using (18) we obtain

$\|U\|\geq 3l(q_{2})/4\geq
3(\lambda(1-23\mu)l(\partial\Pi)-c-2\varepsilon)/4.$

The later inequality contradicts (21) whenever $\mu$ is small enough and
$\rho$ is large enough.

Case 3. Suppose that $l(q_{2})<\|U\|$, i.e., ${\rm\bf Lab\,}(q_{2})$ is a
subword of $U$. Again since $U$ is the shortest word representing $g$ in
$G_{1}$, we have $\|{\rm\bf Lab\,}(q_{2})\|\leq\|Q\|$ for every word $Q$ such
that $Q={\rm\bf Lab\,}(q_{2})$ in $G_{1}$. In particular, for $Q\equiv{\rm\bf
Lab\,}(s_{2}^{-1}us_{1}^{-1})$, we obtain

$\|Q\|\geq\|{\rm\bf
Lab\,}(q_{2})\|\geq\lambda(1-23\mu)l(\partial\Pi)-c-2\varepsilon$ (22)

by (18). On the other hand, we obviously have $\|Q\|\leq
2\varepsilon+23l(\partial\Pi)$, which contradicts (22) if $\mu$ is small
enough and $\rho$ is large enough. ∎

##  7 Words with small cancellations

Throughout the rest of the paper, $G$ denotes a group hyperbolic relative to a
collection of subgroups $\\{H_{\lambda}\\}_{\lambda\in\Lambda}$, $X$ denotes a
finite relative generating set of $G$ with respect to
$\\{H_{\lambda}\\}_{\lambda\in\Lambda}$. As above we set
$\mathcal{A}=X\cup\mathcal{H}.$ Our main goal here is to show that a certain
set of words over the alphabet $\mathcal{A}$ satisfies the small cancellation
conditions described above.

More precisely, suppose that $W$ is a word satisfying the following
conditions.

  1. (W1)

$W\equiv xa_{1}b_{1}\ldots a_{n}b_{n}$ for some $n\geq 1$, where:

  2. (W2)

$x\in X\cup\\{1\\}$;

  3. (W3)

$a_{1},\ldots,a_{n}$ (respectively $b_{1},\ldots,b_{n}$) are elements of
$H_{\alpha}$ (respectively $H_{\beta}$), where $H_{\alpha}\cap
H_{\beta}=\\{1\\}$ ;

  4. (W4)

the elements $a_{1},\ldots,a_{n},b_{1},\ldots,b_{n}$ do not belong to the set

$\mathcal{F}=\mathcal{F}(\varepsilon)=\\{g\in\langle\Omega\rangle\;:\;|g|_{\Omega}\leq
K(32\varepsilon+70)\\},$ (23)

where $\varepsilon$ is some non–negative integer and the set $\Omega$ and the
constant $K$ are provided by Lemma 3.9.

$p_{1}$$p_{3}$$s$$t$$e$$p_{2}$ Figure 2:

Let $\mathcal{SW}$ denote the set of all subwords of cyclic shifts of $W^{\pm
1}$. As in [23], we say that a path $p$ in $\Gamma(G,X\cup\mathcal{H})$ is a
path without backtracking if all components of $p$ are isolated.

######  Lemma 7.1.

Suppose $p$ is a path in $\Gamma(G,X\cup\mathcal{H})$ such that ${\rm\bf
Lab\,}(p)\in\mathcal{SW}$. Then

1) $p$ is a path without backtracking.

2) $p$ is $(1/3,2)$–quasi–geodesic.

###### Proof.

1) Suppose that $p=p_{1}sp_{2}tp_{3}$, where $s$ and $t$ are two connected
components. Passing to another pair of connected components of $p$ if
necessary, we may assume that $p_{2}$ is a path without backtracking. For
definiteness, we also assume that $s$ and $t$ are $H_{\alpha}$–components. Let
$e$ denote a path of length at most $1$ in $\Gamma(G,X\cup\mathcal{H})$
connecting $s_{+}$ to $t_{-}$ and labelled by an element of $H_{\alpha}$ (see
Fig. 2). It follows from our choice of $W$ and the condition $H_{\alpha}\cap
H_{\beta}=\\{1\\}$ that $l(p_{2})\geq 2$. Thus $p_{2}$ contains $m\geq
l(p_{2})/2\geq 1$ $H_{\beta}$–components, say $r_{1},\ldots,r_{m}$, and all
these components are isolated components of the cycle $d=ep_{2}^{-1}$. Let
$g_{1},\ldots,g_{m}$ be elements of $G$ represented by the labels of
$r_{1},\ldots,r_{m}$. By Lemma 3.9, $g_{i}\in\langle\Omega\rangle$,
$i=1,\ldots,m$. According to (W4), we have

$\sum\limits_{i=1}^{m}|g_{i}|_{\Omega}\geq 70Km\geq
35Kl(p_{2})>K(l(p_{2})+1)=Kl(d).$

This contradicts Lemma 3.9.

2) Since the set $\mathcal{SW}$ is closed under taking subwords, it suffices
to show that $dist_{X\cup\mathcal{H}}(p_{-},p_{+})\geq l(p)/3-2$. In case
$l(p)\leq 6$ this is obvious, so we assume $l(p)>6$. Suppose that
$dist_{X\cup\mathcal{H}}(p_{-},p_{+})<l(p)/3-2$. Let $c$ denote a geodesic in
$\Gamma(G,X\cup\mathcal{H})$ such that $c_{-}=p_{-}$ and $c_{+}=p_{+}$. Since
$p$ is a path without backtracking, any $H_{\alpha}$–component of $p$ is
connected to at most one $H_{\alpha}$–component of $c$. Obviously the path $p$
contains at least $l(p)/2-1$ $H_{\alpha}$–components. Therefore, at least

$k=l(p)/2-1-l(c)>l(p)/2-1-(l(p)/3-2)>l(p)/6>1$

of them are isolated $H_{\alpha}$–components of the cycle $pc^{-1}$. Let
$f_{1},\ldots,f_{k}$ be the elements of $G$ represented by these components.
Then as above we have $f_{i}\in\langle\Omega\rangle$, $i=1,\ldots,k$. By (W4),
we obtain

$\sum\limits_{i=1}^{k}|f_{i}|_{\Omega}\geq 70Kk>2Kl(p)>Kl(pc^{-1}).$

This leads to a contradiction again. ∎

######  Lemma 7.2.

Suppose that $upv^{-1}q^{-1}$ is an an arbitrary quadrangle in
$\Gamma(G,X\cup\mathcal{H})$ satisfying the following conditions:

(a) ${\rm\bf Lab\,}(p)\equiv{\rm\bf Lab\,}(q^{-1})\in\mathcal{SW}$;

(b) $\max\\{l(u),\,l(v)\\}\leq\varepsilon$;

(c) $l(p)=l(q)\geq 6\varepsilon+22$.

Then the paths $p$ and $q$ have a common edge.

The proof of Lemma 7.2 is based on the following two results. (We keep the
notation of Lemma 7.2 there.)

######  Lemma 7.3.

Let us divide $p$ into three parts $p=p_{1}p_{0}p_{2}$ such that

$l(p_{1})=l(p_{2})=3\varepsilon+6.$ (24)

Suppose that $s$ is a component of $p_{0}$. Then $s$ can not be connected to a
component of paths $u$ or $v$.

###### Proof.

Suppose that a component $s$ of $p_{0}$ is connected to a component $t$ of
$u$. Then $dist_{X\cup\mathcal{H}}(s_{+},t_{+})\leq 1$. Recall that the
segment $[p_{-},s_{+}]$ of $p$ is $(1/3,2)$–quasi–geodesic by Lemma 7.1.
Hence,

$\begin{array}[]{rl}l(p_{1})&<l([p_{-},s_{+}])\leq
3(dist_{X\cup\mathcal{H}}(p_{-},s_{+})+2)\\\ &\\\ &\leq
3(dist_{X\cup\mathcal{H}}(p_{-},t_{+})+dist_{X\cup\mathcal{H}}(t_{+},s_{+})+2)\\\
&\\\ &\leq 3(l(u)-1+1+2)\leq 3\varepsilon+6.\end{array}$

However, this contradicts (24). Similarly $s$ can not be connected to a
component of $v$. ∎

######  Lemma 7.4.

Let $s_{1},\ldots,s_{k}$ be conseqcutive components of $p_{0}$. Then $q$ can
be decomposed as $q=q_{1}t_{1}\ldots q_{k}t_{k}q_{k+1}$, where

1) $t_{i}$ is a component of $q$ connected to $s_{i}$, $i=1,\ldots,k$;

2) $q_{i}$ contains no components for $i=2,\ldots,k$, i.e. either ${\rm\bf
Lab\,}(q_{i})\equiv x$ or $q_{i}$ is trivial.

###### Proof.

To prove the first assertion of the lemma we proceed by induction. First let
us show that $s_{1}$ is not isolated in $d=upv^{-1}q^{-1}$.

Indeed assume $s_{1}$ is isolated in $d$. Suppose for definiteness that
$s_{1}$ is an $H_{\alpha}$–component. We consider the maximal subpath $s$ of
$p_{0}$ such that $s$ contains $s_{1}$ and all $H_{\alpha}$–components of $s$
are isolated in $d$. By maximality of $s$, either $s_{-}=(p_{0})-$, or
$s_{-}=r_{+}$ for a certain $H_{\alpha}$–component $r$ of $p_{0}$ such that
$r$ is not isolated in $d$. (According to Lemma 7.3 this means that $r$ is
connected to an $H_{\alpha}$–component of $q$.) In the first case we denote by
$f_{1}$ the path $up_{1}$. In the second case, let $f_{1}$ be a path of length
$\leq 1$ that connects an $H_{\alpha}$–component of $q$ to $r$. In both cases
we have $l(f_{1})\leq 4\varepsilon+6$. It follows from the choice of $s$ that
no $H_{\alpha}$–component of $s$ is connected to an $H_{\alpha}$–component of
$f_{1}$. Similarly we construct a path $f_{2}$ such that $(f_{2})_{-}\in q$,
$(f_{2})_{+}=s_{+}$, $l(f_{2})\leq 4\varepsilon+6$, and no
$H_{\alpha}$–component of $s$ is connected to an $H_{\alpha}$–component of
$f_{2}$.

Clearly all $H_{\alpha}$–components of $s$ are isolated in the cycle
$c=f_{1}sf_{2}^{-1}[(f_{2})_{-},(f_{1})_{-}]$, where
$[(f_{2})_{-},(f_{1})_{-}]$ is a segment of $q^{\pm 1}$. We have

$dist_{X\cup\mathcal{H}}((f_{1})_{-},(f_{2})_{-})\leq
l(f_{1})+l(s)+l(f_{2})\leq 8\varepsilon+12+l(s).$ (25)

Consequently,

$l([(f_{1})_{-},(f_{2})_{-}])\leq
3dist_{X\cup\mathcal{H}}((f_{1})_{-},(f_{2})_{-})+2)\leq
24\varepsilon+42+3l(s).$ (26)

Finally,

$l(c)\leq l(f_{1})+l(s)+l(f_{2})+l([(f_{1})_{-},(f_{2})_{-}])\leq
32\varepsilon+54+4l(s).$ (27)

Let $g_{1},\ldots,g_{m}$ denote the elements represented by
$H_{\alpha}$–components of $s$. Note that $l(s)\leq 2m+2$. Applying Lemma 3.9,
we obtain $g_{i}\in\langle\Omega\rangle$, $i=1,\ldots,m$, and

$\sum\limits_{i=1}^{m}|g_{i}|_{\Omega}\leq Kl(c)\leq
K(32\varepsilon+54+4l(s))\leq K(32\varepsilon+62+8m).$ (28)

Therefore, at least one of the elements $g_{1},\ldots,g_{m}$ has length at
least

$\frac{1}{m}K(32\varepsilon+62+8m)\leq K(32\varepsilon+70)$ (29)

that contradicts (W4). Thus $s_{1}$ is not isolated in $d$. By Lemma 7.3 this
means that $s_{1}$ is connected to an $H_{\alpha}$–component $t_{1}$ of $q$.

Now assume that we have already found components $t_{1},\ldots,t_{i}$ of $q$,
$1\leq i<k$, that are connected to $s_{1},\ldots,s_{i}$ respectively. The
inductive step is similar to the above considerations. For definiteness, we
assume that $s_{i}$ is an $H_{\alpha}$–component. Then $s_{i+1}$ is an
$H_{\beta}$–component by the choice of $W$. We denote by $f_{1}$ a path of
length $\leq 1$ labelled by an element of $H_{\alpha}$ that connects
$(t_{i})_{+}$ to $(s_{i})_{+}$ (Fig. 3). If $s_{i+1}$ is isolated in the cycle
$c=f_{1}[(s_{i})_{+},p_{+}]v^{-1}[q_{+},(t_{i})_{+}]$, we denote by $s$ the
maximal initial subpath of the segment $[(s_{i})_{+},(p_{0})_{+}]$ of $p_{0}$
such that $s$ contains $s_{i+1}$ and all $H_{\alpha}$–components of $s$ are
isolated in $c$. As above, we can find a path $f_{2}$ such that
$(f_{2})_{-}\in q$, $(f_{2})_{+}=s_{+}$, $l(f_{2})\leq 4\varepsilon+6$, and no
$H_{\alpha}$–component of $s$ is connected to an $H_{\alpha}$–component of
$f_{2}$. The inequalities (25)–(29) remain valid and we arrive at a
contradiction in the same way. Thus $s_{i+1}$ is not isolated in $c$, i.e., it
is connected to a component $t_{i+1}$ of the segment $[(t_{i})_{+},q_{+}]$ of
$q$. This completes the inductive step.

$u$$v$$q$$p$$t_{i}$$s_{i}$$s_{-}$$s_{+}$$s_{i+1}$$f_{1}$$f_{2}$$c$ Figure 3:

Let us prove the second assertion of the lemma. Suppose that $s_{i},s_{i+1}$
are two subsequent components of $p_{0}$ and such that $q_{i+1}$ contains at
least one component (say, an $H_{\alpha}$–component). As above for
definiteness we assume that $s_{i}$ (respectively $s_{i+1}$) is an
$H_{\alpha}$–component (respectively $H_{\beta}$–component). Let $e_{1}$,
$e_{2}$ be the paths of lengths $\leq 1$ labelled by elements of $H_{\alpha}$
and $H_{\beta}$ respectively such that $(e_{1})_{+}=(s_{i})_{+}$,
$(e_{1})_{-}=(q_{i+1})_{-}$, $(e_{2})_{+}=(s_{i+1})_{-}$,
$(e_{2})_{-}=(q_{i+1})_{+}$ (see Fig. 4). As $q$ is a path without
backtracking, each $H_{\alpha}$–component of $q_{i+1}$ is isolated in the
cycle $e=q_{i+1}e_{2}[(s_{i+1})_{-},(s_{i})_{+}]e_{1}^{-1}$, where
$[(s_{i+1})_{-},(s_{i})_{+}]$ is a segment of $p^{-1}$. Notice that
$l([(s_{i+1})_{-},(s_{i})_{+}])\leq 1$ as $s_{i}$ and $s_{i+1}$ are subsequent
components of $p_{0}$. We denote by $f_{1},\ldots,f_{m}$ the elements
represented by $H_{\alpha}$–components of $q_{i+1}$. By Lemma 3.9, we have
$f_{j}\in\langle\Omega\rangle$, $j=1,\ldots,m$, and

$\sum\limits_{j=1}^{m}|f_{j}|_{\Omega}\leq Kl(e)\leq K(3+l(q_{i+1}))\leq
K(3+2m+2)\leq 7mK.$

This contradicts (W4) again. ∎

$u$$v$$q$$p$$t_{i}$$s_{i}$$s_{i+1}$$t_{i+1}$$q_{i+1}$$e_{1}$$e_{2}$$e$ Figure
4:

###### Proof of Lemma 7.2.

We keep the notation introduced in Lemma 5.3 and Lemma 5.4. Let also

$p_{0}=p_{1}s_{1}\ldots p_{k}s_{k}p_{k+1}$

for some (may be trivial) subpaths $p_{1},\ldots p_{k+1}$ of $p_{0}$.

According to (24) and condition c) of Lemma 7.2, we have

$l(p_{0})\geq 6\varepsilon+22-l(p_{1})-l(p_{2})\geq 10.$

Since $s_{1},\ldots,s_{k}$ are subsequent components and ${\rm\bf
Lab\,}(p_{0})$ is a subword of a cyclic shift of $W^{\pm 1}$, at most one of
the paths $p_{2},\ldots,p_{k}$ is non–trivial. Therefore, there are at least
$5$ subsequent components $s_{i},\ldots,s_{i+4}$, such that
$p_{i+1},\ldots,p_{i+4}$ are trivial. Without loss of generality we may assume
$i=1$. Similarly, by Lemma 7.4, we can find at least three subsequent
components among $t_{1},\ldots t_{5}$, say $t_{1},t_{2},t_{3}$, such that
$(t_{1})_{+}=(t_{2})_{-}$ and $(t_{2})_{+}=(t_{3})_{-}$. Let $w$ be an element
represented by the label of any path that goes from $(t_{1})_{+}=(t_{2})_{-}$
to $(s_{1})_{+}=(s_{2})_{-}$. For definiteness we assume that $t_{1}$ and
$s_{1}$ are $H_{\alpha}$–components. Since $t_{1}$ and $s_{1}$ are connected,
we have $w\in H_{\alpha}$. On the other hand, the $H_{\beta}$–components
$t_{2}$ and $s_{2}$ are also connected. Hence $w\in H_{\beta}$. Thus $w\in
H_{\alpha}\cap H_{\beta}=\\{1\\}$, i.e. the vertices $(t_{2})_{-}$ and
$(s_{2})_{-}$ coincide. Similarly the vertices $(t_{2})_{+}$ and $(s_{2})_{+}$
coincide. In particular, $t_{2}$ and $s_{2}$ are edges labelled by the same
element of $H_{\beta}$, i.e., $t_{2}$ and $s_{2}$ coincide. ∎

Now we are ready to prove the main result of this section.

######  Theorem 7.5.

Suppose that $W$ is a word in $\mathcal{A}$ satisfying the conditions
(W1)–(W4) and, in addition, $a_{i}\neq a_{j}^{\pm 1}$, $b_{i}\neq b_{j}^{\pm
1}$ whenever $i\neq j$ and $a_{i}\neq a_{i}^{-1}$, $b_{i}\neq b_{i}^{-1}$,
$i,j\in\\{1,\ldots,n\\}$. Then the set $\mathcal{W}$ of all cyclic shifts of
$W^{\pm 1}$ satisfies the
$C_{1}(\varepsilon,\frac{3\varepsilon+11}{n},\frac{1}{3},2,2n+1)$ small
cancellation condition.

###### Proof.

The first two conditions from Definition 4.2 follow from the choice of $W$ and
Lemma 7.1. Suppose that $U$ is an $\varepsilon$–piece of a word
$R\in\mathcal{W}$. Assume that $\max\\{\|U\|,\|U^{\prime}\|\\}\geq\mu\|R\|$
for $\mu=\frac{3\varepsilon+11}{n}$, that is,

$\max\\{\|U\|,\|U^{\prime}\|\\}\geq\frac{3\varepsilon+11}{n}(2n+1)>6\varepsilon+22.$

(Here and below we use the notation of Definitions 4.2 and 4.1.) Without loss
of generality we may assume that $\|U\|\geq 6\varepsilon+22$. By the
definition of an $\varepsilon$–piece, there is a quadrangle $upv^{-1}q^{-1}$
in $\Gamma(G,X\cup\mathcal{H})$ satisfying conditions (a)–(c) of Lemma 7.2 and
such that labels of $p$ and $q$ are $U$ and $U^{\prime}$ respectively.

Let $e$ be the common edge of $p$ and $q$. Then we have

$R\equiv U_{1}{\rm\bf Lab\,}(e)U_{2}V$

and

$R^{\prime}\equiv U_{1}^{\prime}{\rm\bf Lab\,}(e)U_{2}^{\prime}V^{\prime},$

where $U_{1}{\rm\bf Lab\,}(e)U_{2}\equiv U$ and $U^{\prime}\equiv
U_{1}^{\prime}{\rm\bf Lab\,}(e)U_{2}^{\prime}$. Since ${\rm\bf Lab\,}(e)$
appears in $W^{\pm 1}$ only once, $R$, $R^{\prime}$ are cyclic shifts of the
same word $W^{\pm 1}$ and

$U_{2}VU_{1}\equiv U_{2}^{\prime}V^{\prime}U_{1}^{\prime}.$

Note also that

$Y=U_{1}^{\prime}U_{1}^{-1}$

in $G$ as $Y^{-1}U_{1}^{\prime}U_{1}^{-1}$ is a label of a cycle in
$\Gamma(G,X\cup\mathcal{H})$. Therefore, the following equalities hold in the
group G:

$YRY^{-1}=U_{1}^{\prime}U_{1}^{-1}U_{1}{\rm\bf
Lab\,}(e)U_{2}VU_{1}(U_{1}^{\prime})^{-1}=U_{1}^{\prime}{\rm\bf
Lab\,}(e)U_{2}^{\prime}V^{\prime}U_{1}^{\prime}(U_{1}^{\prime})^{-1}=R^{\prime}$

that contradicts the third condition from Definition 4.2.

Similarly, if $U$ is an $\varepsilon^{\prime}$–piece, then $R\equiv
UVU^{\prime}V^{\prime}$ for some $U,U^{\prime},V,V^{\prime}$, where both
subwords $U$ and $U^{\prime}$ contain a certain letter from
$X\cup\mathcal{H}$. In this case we arrive at a contradiction again as any
letter $a\in X\cup\mathcal{H}$ appears in $R$ only once, and if $a$ appears in
$R$, then $a^{-1}$ does not. ∎

Finally we note that the condition $x\in X\cup\\{1\\}$ in Theorem 7.5 is
imposed by some technical reasons and it is not restrictive since we can
always add any element $x\in G$ to the set $X$ without violating relative
hyperbolicity.

##  8 Suitable subgroups and quotients

Throughout this section, we keep the assumption that $G$ is hyperbolic
relative to a collection of subgroups $\\{H_{\lambda}\\}_{\lambda\in\Lambda}$.
The proof of Lemma 2.3 is based on the following auxiliary result.

######  Lemma 8.1.

Suppose that for some $\lambda,\mu\in\Lambda$, $\lambda\neq\mu$, $H_{\lambda}$
and $H_{\mu}$ contain elements of infinite order and $H_{\lambda}\cap
H_{\mu}=\\{1\\}$. Then there are $f\in H_{\lambda}$, $g\in H_{\mu}$ such that
$fg$ is a hyperbolic element of infinite order and $E_{G}(fg)=\langle
fg\rangle$.

###### Proof.

We set $\varepsilon=2(\kappa+\delta)$, where $\kappa=\kappa(\delta,1/3,2)$ is
the constant provided by Lemma 3.1 and $\delta$ is the hyperbolicity constant
of $\Gamma(G,X\cup\mathcal{H})$. It is convenient to assume that
$\kappa,\delta\in\mathbb{N}$.

Let $\mathcal{F}=\mathcal{F}(\varepsilon)$ be the set defined by (23). Since
$\Omega$ is finite, $\mathcal{F}$ is finite, and hence there are elements
$f\in H_{\lambda}\setminus\mathcal{F}$ and $g\in H_{\mu}\setminus\mathcal{F}$
of infinite order. In particular,

$f^{2}\neq 1,\;\;\;g^{2}\neq 1.$ (30)

Note that the word $W=(fg)^{m}$ satisfies conditions (W1)–(W4). (Here $f$ and
$g$ are regarded as letters of $\mathcal{H}$.) By Lemma 7.1, for any
$m\in\mathbb{N}$, the word $(fg)^{m}$ is $(1/3,2)$–quasi–geodesic in
$\Gamma(G,X\cup\mathcal{H})$. In particular, $fg$ is a hyperbolic element.
Indeed otherwise the length $|(fg)^{m}|_{X\cup\mathcal{H}}$ would be bounded
uniformly on $m$.

Now suppose that $a\in E_{G}(fg)$. Then by Theorem 2.1,
$a(fg)^{m}a^{-1}=(fg)^{\pm m}$ for some $m\in\mathbb{N}$. Passing to a
multiple of $m$ if necessary, we may assume that

$m\geq 3|a|_{X\cup\mathcal{H}}+12(\kappa+\delta)+20.$ (31)

Let $a_{1}b_{1}a_{2}b_{2}$ be a quadrangle in $\Gamma(G,X\cup\mathcal{H})$
such that $a_{1}$, $a_{2}$ are geodesic, the labels ${\rm\bf
Lab\,}(a_{1})={\rm\bf Lab\,}(a_{2}^{-1})$ represent $a$ in $G$, and ${\rm\bf
Lab\,}(b_{1})={\rm\bf Lab\,}(b_{2}^{-1})\equiv(fg)^{\pm m}$. Let also
$b_{1}=b_{1}^{\prime}pb_{1}^{\prime\prime}$ (Fig. 5), where

$l(b_{1}^{\prime})=l(b_{1}^{\prime\prime})=3(l(a_{1})+2(\kappa+\delta)+3).$
(32)

As $b_{1}$ is $(1/3,2)$–quasi–geodesic, we have

$dist_{X\cup\mathcal{H}}(p_{\pm},(b_{1})_{\pm})\geq\frac{1}{3}l(b_{1}^{\prime})-2\geq
l(a_{1})+2(\kappa+\delta)+1.$ (33)

Further by Corollary 3.3, there is a point $s\in a_{1}\cup b_{2}\cup a_{2}$
such that $dist_{X\cup\mathcal{H}}(s,p_{-})\leq 2(\kappa+\delta)$. If $s\in
a_{1}$, then we have

$dist_{X\cup\mathcal{H}}(p_{-},b_{-})\leq
dist_{X\cup\mathcal{H}}(p_{-},s)+dist_{X\cup\mathcal{H}}(s,b_{-})\leq
2(\kappa+\delta)+l(a_{1})$

that contradicts (33). For the same reason $s$ can not belong to $a_{2}$. Thus
$s\in b_{2}$. Without loss of generality, we may assume that $s$ is a vertex
of $\Gamma(G,X\cup\mathcal{H})$. Similarly there exists a vertex $t\in b_{2}$
such that $dist_{X\cup\mathcal{H}}(t,p_{+})\leq 2(\kappa+\delta)$. Let $u,v$
be geodesics in $\Gamma(G,X\cup\mathcal{H})$ connecting $s$ to $p_{-}$ and $t$
to $p_{+}$ respectively and let $q$ denote the segment $[s,t]$ of
$b_{2}^{-1}$.

According to (31) and (32), we have

$l(p)\geq 2m-6(l(a_{1})+2(\kappa+\delta)+3)\geq 12(\kappa+\delta)+22.$

Hence we may apply Lemma 7.2 for the quadrangle $upv^{-1}q^{-1}$ and
$\varepsilon=2(\kappa+\delta)$. Thus there exists a common edge $e$ of $p$ and
$q$. In particular, this and (30) imply that $a(fg)^{m}a^{-1}=(fg)^{m}$ (not
$(fg)^{-m}$).

There are two possibilities for labels of the segments $[(b_{1})_{-},e_{-}]$
and $[(b_{2})_{+},e_{-}]$ of $b_{1}$ and $b_{2}^{-1}$ respectively. Namely
both these labels are either of the form $(fg)^{n}$ (possibly for different
$n$) or of the form $(fg)^{k}f$. In both cases $a=(fg)^{l}$ for a certain $l$
as labels of $a_{1}$ and $[(b_{2})_{+},e_{-}][(b_{1})_{-},e_{-}]^{-1}$
represent the same element of $G$. Thus $a\in\langle fg\rangle$ and
$E_{G}(fg)=\langle fg\rangle$. ∎

$b_{1}^{\prime}$$b_{1}^{\prime\prime}$$p_{+}$$p_{-}$$e$$s=q_{-}$$t=q_{+}$$a_{1}$$a_{2}$$v$$u$
Figure 5:

###### Proof of Lemma 2.3.

Let $f_{1},f_{2}\in H^{0}$ be non–commensurable elements of $H$ such that
$E_{G}(f_{1})\cap E_{G}(f_{2})=\\{1\\}$. By Theorem 2.1, $G$ is hyperbolic
relative to the collection $\\{H_{\lambda}\\}_{\lambda\in\Lambda}\cup
E_{G}(f_{1})\cup E_{G}(f_{2})$.

We construct a sequence of desired elements $h_{1},h_{2},\ldots$ by induction.
By Lemma 8.1, there are $f\in E_{G}(f_{1})$, $g\in E_{G}(f_{2})$ such that the
element $h_{1}=fg$ is hyperbolic ( with respect to the collection
$\\{H_{\lambda}\\}_{\lambda\in\Lambda}\cup E_{G}(f_{1})\cup E_{G}(f_{2})$) and
$E_{G}(h_{1})=\langle h_{1}\rangle$. Theorem 2.1 implies that $G$ is
hyperbolic relative to the collection
$\\{H_{\lambda}\\}_{\lambda\in\Lambda}\cup E_{G}(f_{1})\cup E_{G}(f_{2})\cup
E_{G}(h_{1})$. Further we construct a hyperbolic (with respect to
$\\{H_{\lambda}\\}_{\lambda\in\Lambda}\cup E_{G}(f_{1})\cup E_{G}(f_{2})\cup
E_{G}(h_{1})$) element $h_{2}$ as a product of an element of $E_{G}(f_{1})$
and an element of $E_{G}(f_{2})$ as above. As $h_{2}$ is hyperbolic, it is not
commensurable with $h_{1}$. Applying Theorem 2.1 again we join $E_{G}(h_{2})$
to the collection of subgroups with respect to which $G$ is hyperbolic and so
on. Continuing this procedure, we get what we need. ∎

To prove Theorem 2.4, we need the following two observations. The first one is
a particular case of Theorem 2.40 from [23].

######  Lemma 8.2.

Suppose that a group $G$ is hyperbolic relative to a collection of subgroups
$\\{H_{\lambda}\\}_{\lambda\in\Lambda}\cup\\{S_{1},\ldots,S_{m}\\}$, where
$S_{1},\ldots,S_{m}$ are finitely generated and hyperbolic in the ordinary
(non–relative) sense. Then $G$ is hyperbolic relative to
$\\{H_{\lambda}\\}_{\lambda\in\Lambda}$.

The next lemma is a particular case of Theorem 1.4 from [23]

######  Lemma 8.3.

Suppose that a group $G$ is hyperbolic relative to a collection of subgroups
$\\{H_{\lambda}\\}_{\lambda\in\Lambda}$. Then

  1. (a)

For any $g\in G$ and any $\lambda,\mu\in\Lambda$, $\lambda\neq\mu$, the
intersection $H_{\lambda}^{g}\cap H_{\mu}$ is finite.

  2. (b)

For any $\lambda\in\Lambda$ and any $g\notin H_{\lambda}$, the intersection
$H^{g}_{\lambda}\cap H_{\lambda}$ is finite.

###### Proof of Theorem 2.4.

Obviously it suffices to deal with the case $m=1$. The general case will
follow if we apply the theorem for $m=1$ several times.

Let $t\in G$ be an arbitrary element. Passing to a new relative generating set
$X^{\prime}=X\cup\\{t\\}$ if necessary, we may assume that $t\in X$. By Lemma
2.3 there are non–commensurable elements $h_{1},h_{2}\in H^{0}$ such that
$E_{G}(h_{1})$ and $E_{G}(h_{2})$ are cyclic. According to Theorem 2.1, $G$ is
hyperbolic relative to $\\{H_{\lambda}\\}_{\lambda\in\Lambda}\cup
E_{G}(h_{1})\cup E_{G}(h_{2})$.

Let $\mu,\varepsilon,\rho$ be constants that satisfy the conclusions of Lemma
5.1 and Lemma 6.3 for $\lambda=1/3$, $c=2$, $N=1$. By Theorem 7.5, there are
$n$ and $m_{1},\ldots,m_{n}$ such that the set $\mathcal{R}$ of all cyclic
shifts and their inverses of the word

$R\equiv th_{1}^{m_{1}}h_{2}^{m_{1}}\ldots h_{1}^{m_{n}}h_{2}^{m_{n}}$

in the alphabet
$\mathcal{A}=X\cup\mathcal{H}\cup(E_{G}(h_{1})\setminus\\{1\\})\cup(E_{G}(h_{2})\setminus\\{1\\})$
satisfies the $C(1/3,2,\varepsilon,\mu,\rho)$–condition (here $h_{i}^{m_{j}}$
is regarded as a letter in $E_{G}(h_{i})\setminus\\{1\\}$, $i=1,2$,
$j=1,\ldots,n$). Indeed it suffices to chose large enough $n$ and
$m_{1},\ldots,m_{n}$ satisfying $m_{i}\neq\pm m_{j}$ whenever $i\neq j$. Let
$G_{1}$ be the quotient of $G$ obtained by imposing the relation $R=1$ and
$\eta$ the corresponding natural homomorphism.

By Lemma 5.1, $G_{1}$ is hyperbolic relative to the images of
$H_{\lambda},\lambda\in\Lambda$ and $E_{G}(h_{1})$, $E_{G}(h_{2})$. As any
elementary group is hyperbolic, $G_{1}$ is also hyperbolic relative to
$\\{\eta(H_{\lambda}\\}_{\lambda\in\Lambda}\\}$ according to Lemma 8.2. The
inclusion $\eta(t)\in\eta(H)$ follows immediately from the equality $R=1$ in
$G_{1}$. The third assertion of the theorem follows from Lemma 5.1 b) as any
element from the union $\bigcup\limits_{\lambda\in\Lambda}H_{\lambda}$ has
length $1$.

Similarly as $\eta$ is injective on $E_{G}(h_{1})\cup E_{G}(h_{2})$,
$\eta(h_{1})$ and $\eta(h_{2})$ are elements of infinite order. Note also that
$\eta(h_{1})$ and $\eta(h_{2})$ are not commensurable in $G_{1}$. Indeed
otherwise the intersection
$\big{(}\eta(E_{G}(h_{1}))\big{)}^{g}\cap\eta(E_{G}(h_{2}))$ is infinite for
some $g\in G$ contradictory the first assertion of Lemma 8.3. Assume now that
$g\in E_{G_{1}}(\eta(h_{1}))$, where $E_{G_{1}}(\eta(h_{1}))$ is the maximal
elementary subgroup of $G_{1}$ containing $\eta(h_{1})$. By the first
assertion of Theorem 2.1, $\big{(}\eta(h_{1}^{m})\big{)}^{g}=\eta(h_{1}^{\pm
m})$ for a certain $m\neq 0$. Therefore,
$\big{(}\eta(E_{G}(h_{1}))\big{)}^{g}\cap\eta(E_{G}(h_{1}))$ contains
$\eta(h_{1}^{m})$ and in particular this intersection is infinite. By the
second assertion of Lemma 8.3, this means that $g\in\eta(E_{G}(h_{1}))$. Thus
we proved that $\eta(E_{G}(h_{1}))=E_{G_{1}}(\eta(h_{1}))$. The same is true
for $h_{2}$. Finally, using injectivity of $\eta$ on $E_{G}(h_{1})\cup
E_{G}(h_{2})$ again, we obtain

$E_{G_{1}}(\eta(h_{1}))\cap
E_{G_{1}}(\eta(h_{2}))=\eta(E_{G}(h_{1}))\cap\eta(E_{G}(h_{2}))=\eta\big{(}E_{G}(h_{1})\cap
E_{G}(h_{2})\big{)}=\\{1\\}.$

This means that the image of $H$ is a suitable subgroup of $G_{1}$. To
complete the proof it remains to note that the last assertion of the theorem
follows from Lemma 6.3. ∎

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Stevenson Center 1326, Department of Mathematics, Vanderbilt University
Nashville, TN 37240, USA

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