instead of $\mathsf{Y}_{1}$ and $\mathsf{M}_{2}^{\prime}$ instead of
$\mathsf{M}_{2}$ we find $\mathsf{N}_{2}$ in $\mathsf{Y}_{2}$ disjoint from
$\mathsf{M}_{1}$ such that $\mu_{\mathrm{f}}(\mathsf{N}_{2})>5.7\omega$ and
$\mathsf{N}_{2}\sim\mathsf{N}^{\pm 1}$. Then we can take
$\mathsf{N}^{\prime}=\mathsf{N}_{1}\cup\mathsf{N}_{2}$ by Corollary 9.24(i),
LABEL:coi:no-inverse-compatibility-alpha-it@m.

If $\mu_{\mathrm{f}}(\mathsf{N})\geq 4\lambda+9\omega$ then
$\mu_{\mathrm{f}}(\mathsf{N}^{\prime})>2\lambda+5.6\omega$ and using
Propositions 8.11β and 8.10β we conclude that $\mathsf{M}_{1}$ and
$\mathsf{M}_{2}$ cannot cover $\mathsf{N}^{\prime}$ together, i.e.
$\mathsf{M}_{1}\ll\mathsf{M}_{2}$.

In case (**) we already have a loop
$\mathsf{X}^{-1}\mathsf{u}\mathsf{Y}\mathsf{v}$ with bridges $\mathsf{u}$ and
$\mathsf{v}$ of rank $\beta$. We lift it to $\Gamma_{\beta}$ and then apply
Lemma 10.20β to see that the lift of $\mathsf{N}$ is independent of the lifts
of $\mathsf{u}$ and $\mathsf{v}$. Then application of Proposition 10.6β gives
the required $\mathsf{N}^{\prime}$.

(ii): An easy analysis with a help of Propositions 9.24(ii) and 8.10β shows
that it is enough to prove the following: Let $\mathsf{X}$ and $\mathsf{Y}$ be
reduced paths in $\Gamma_{\alpha}$. Let $\mathsf{K}_{i}$ $(i=1,2,3)$ be
fragments of rank $\beta$ in $\mathsf{X}$, $\mathsf{M}_{i}$ $(i=1,2,3)$ be
fragments of rank $\beta$ in $\mathsf{Y}$,
$\mu_{\mathrm{f}}(\mathsf{K}_{i}),\mu_{\mathrm{f}}(\mathsf{M}_{i})\geq\lambda+9.1\omega$,
$\mathsf{K}_{i}\sim\mathsf{M}_{i}^{\pm 1}$ for all $i$ and
$\mathsf{K}_{i}\not\sim\mathsf{K}_{j}$ for $i\neq j$. If
$\mathsf{K}_{1}<\mathsf{K}_{2}<\mathsf{K}_{3}$ and
$\mathsf{M}_{1}<\mathsf{M}_{3}$ then
$\mathsf{M}_{1}<\mathsf{M}_{2}<\mathsf{M}_{3}$.

Assume that this is not the case, that is, we have
$\mathsf{K}_{1}<\mathsf{K}_{2}<\mathsf{K}_{3}$,
$\mathsf{M}_{1}<\mathsf{M}_{3}$ and either $\mathsf{M}_{2}<\mathsf{M}_{1}$ or
$\mathsf{M}_{3}<\mathsf{M}_{2}$. By (i), there exists a fragment $\mathsf{N}$
of rank $\alpha$ in $\mathsf{Y}$ such that $\mathsf{K}_{2}\sim\mathsf{N}^{\pm
1}$ and $\mathsf{M}_{1}<\mathsf{N}<\mathsf{M}_{3}$. Then by Propositions
9.24(i) and 8.10β we obtain $\mathsf{M}_{1}\sim\mathsf{N}$ or
$\mathsf{M}_{3}\sim\mathsf{N}$, a contradiction. ∎

######  10.24 Proposition.

Let $X$ and $Y$ be words strongly cyclically reduced in $G_{\alpha}$,
representing conjugate elements of $G_{\alpha}$. Let $\bar{\mathsf{X}}$ and
$\bar{\mathsf{Y}}$ be lines in $\Gamma_{\alpha}$ representing the conjugacy
relation. Let $1\leq\beta\leq\alpha$. Assume that at least one of the words
$X$ or $Y$ has the property that no its cyclic shift contains a fragment $K$
of rank $\gamma$ with $\mu_{\mathrm{f}}(K)>\xi_{0}$ and
$\beta<\gamma\leq\alpha$. Let
$\bar{\mathsf{X}}=\dots\mathsf{X}_{-1}\mathsf{X}_{0}\mathsf{X}_{1}\dots$ and
$\bar{\mathsf{Y}}=\dots\mathsf{Y}_{-1}\mathsf{Y}_{0}\mathsf{Y}_{1}\dots$ be
lines in $\Gamma_{\alpha}$ representing the conjugacy relation.

  1. (i)

Then for any fragment $\mathsf{K}$ of rank $\beta$ in $\bar{\mathsf{X}}$ with
$\mu_{\mathrm{f}}(\mathsf{K})\geq 2\lambda+9.1\omega$ there exists a fragment
$\mathsf{M}$ of rank $\beta$ in $\bar{\mathsf{Y}}$ such that
$\mathsf{M}\sim\mathsf{K}^{\pm 1}$ and

$\mu_{\mathrm{f}}(\mathsf{M})\geq\min\\{\mu_{\mathrm{f}}(\mathsf{K})-2\lambda-3.4\omega,\
\xi_{0}\\}$

  2. (ii)

If $X$ and $Y$ are strongly cyclically reduced in $G_{\alpha}$ then the
correspondence between fragments of rank $\beta$ in $\bar{\mathsf{X}}$ and in
$\bar{\mathsf{Y}}$ preserves the ordering in the following sense: if
$\mathsf{K}_{i}$ $(i=1,2)$ are fragments of rank $\beta$ in
$\bar{\mathsf{X}}$, $\mathsf{M}_{i}$ $(i=1,2)$ are fragments of rank $\beta$
in $\bar{\mathsf{Y}}$,
$\mu_{\mathrm{f}}(\mathsf{K}_{i}),\mu_{\mathrm{f}}(\mathsf{M}_{i})\geq
2\lambda+9.1\omega$, $\mathsf{K}_{i}\sim\mathsf{M}_{i}^{\pm 1}$ $(i=1,2)$ and
$\mathsf{K}_{1}\not\sim\mathsf{K}_{2}$. Then $\mathsf{K}_{1}<\mathsf{K}_{2}$
if and only if $\mathsf{M}_{1}<\mathsf{M}_{2}$.

Furthermore, the statement of the proposition is true also in the case
$\beta=0$ if we drop all conditions of the form
$\mu_{\mathrm{f}}(\cdot)\geq\dots$ for fragments of rank $\beta$.

###### Proof.

By Proposition 9.17 every subpath of $\bar{\mathsf{X}}$ can be extended to be
close in rank $\beta$ to a subpath of $\bar{\mathsf{Y}}$. Then (i) follows
from Proposition 8.16(ii) and Proposition 10.23(i) with
$\mathsf{K}_{1}=s_{X,\bar{\mathsf{X}}}^{-1}\mathsf{K}$ and
$\mathsf{K}_{2}=s_{X,\bar{\mathsf{X}}}\mathsf{K}$. Statement (ii) follows by
Proposition 10.23(ii). In the case $\beta=0$ the statement becomes trivial
after application of Proposition 9.17. ∎

##  11\. Reduced representatives

The main goal of this section is to prove that any element of $G_{\alpha}$ can
be represented by a reduced word and to prove a cyclic analog of this
statement (Proposition 11.5).

######  11.1 Proposition (reduced representative).

Every element of $G_{\alpha}$ can be represented by a reduced in $G_{\alpha}$
word which contains no fragments $F$ of rank $1\leq\beta\leq\alpha$ with
$\mu_{\mathrm{f}}(\mathsf{F})\geq\frac{1}{2}+2\lambda+15\omega$.

######  11.2 Lemma.

Let $m\geq 3$ and
$\mathsf{X}^{-1}*\mathsf{Y}_{1}*\mathsf{Y}_{2}*\dots*\mathsf{Y}_{m}*$ be a
coarse $(m+1)$-gon in $\Gamma_{\alpha-1}$. Assume that there are indices
$1\leq t_{1}<t_{2}<\dots<t_{k}\leq m$ $(k\geq 1)$ such that

$t_{1}\leq 3,\quad t_{k}\geq m-2,\quad t_{j}-t_{j-1}\leq 2\text{ for all }j$

and

$|\mathsf{Y}_{t_{j}}|_{\alpha-1}>4\eta\quad\text{for all }j.$

Assume further that there are no close vertices in each of the pairs
$(\mathsf{Y}_{i},\mathsf{Y}_{i+1})$, $(\mathsf{Y}_{1},\mathsf{Y}_{t_{1}})$,
$(\mathsf{Y}_{t_{j}},\mathsf{Y}_{t_{j}+1})$,
$(\mathsf{Y}_{t_{k}},\mathsf{Y}_{m})$ except appropriate endpoints (i.e.
except $\tau(\mathsf{Y}_{i})$ and $\iota(\mathsf{Y}_{i+1})$). Then each of the
paths $\mathsf{Y}_{t_{j}}$ has a vertex close to a vertex $\mathsf{a}_{j}$ on
$\mathsf{X}$ and these vertices $\mathsf{a}_{j}$ are in $\mathsf{X}$ in the
(non-strict) order from start to end.

###### Proof.

We first claim that there are no close vertices in pairs
$(\mathsf{Y}_{i},\mathsf{Y}_{j})$ for $j-i>1$. Assume there are. We choose
such a pair with minimal possible $j-i$. Then an ending segment
$\mathsf{Y}_{i}^{\prime}$ of $\mathsf{Y}_{i}$, paths $\mathsf{Y}_{i+1}$,
$\dots$, $\mathsf{Y}_{j-1}$ and a starting segment $\mathsf{Y}_{j}^{\prime}$
of $\mathsf{Y}_{j}$ form a coarse $r$-gon with $r=j-i+1\geq 3$. Applying
Proposition 9.18α-1 we get

$\sum_{k=i+1}^{j-1}|\mathsf{Y}_{i}|_{\alpha-1}\leq(r-2)\eta.$

On the other hand, it follows from the hypothesis of the lemma that there are
at least $\min(1,\frac{1}{2}(r-3))$ paths $\mathsf{Y}_{t_{k}}$ among
$\mathsf{Y}_{i+1}$, $\dots$, $\mathsf{Y}_{j-1}$ and hence

$\sum_{k=i+1}^{j-1}|\mathsf{Y}_{i}|_{\alpha-1}>4\eta\min\left(1,\frac{1}{2}(r-3)\right).$

We get a contradiction since the right hand side of the inequality is at least
$(r-2)\eta$. This proves the claim.

Shortening if necessary $\mathsf{Y}_{1}$ and $\mathsf{X}$ we can assume that
there is no pair of close vertices on $\mathsf{Y}_{1}$ and $\mathsf{X}$ other
that $(\iota(\mathsf{Y}_{1}),\iota(\mathsf{X}))$. Similarly, we can assume
that there is no pair of close vertices on $\mathsf{Y}_{m}$ and $\mathsf{X}$
other than $(\tau(\mathsf{Y}_{m}),\tau(\mathsf{X}))$. Now we claim that there
is a pair of close vertices on $\mathsf{Y}_{i}$ and $\mathsf{X}$ for some
$2\leq i\leq m-1$. Indeed, otherwise we can apply Proposition 9.18α-1 to the
whole coarse $(m+1)$-gon
$\mathsf{X}^{-1}*\mathsf{Y}_{1}*\mathsf{Y}_{2}*\dots*\mathsf{Y}_{m}*$ and
obtain a contradiction since $4k\eta\geq(m-1)\eta$.

Let $(\mathsf{b},\mathsf{c})$ be a pair of close vertices on $\mathsf{X}$ and
$\mathsf{Y}_{i_{0}}$ where $2\leq i_{0}\leq m-1$. Let $\mathsf{b}$ divide
$\mathsf{X}$ as $\mathsf{X}_{1}\mathsf{X}_{2}$ and $\mathsf{c}$ divide
$\mathsf{Y}_{i_{0}}$ as $\mathsf{Z}_{1}\mathsf{Z}_{2}$ If there is at least
one index $t_{j}$ in the interval $2\leq t_{j}\leq i_{0}-1$ then the
conditions of the lemma are satisfied for the coarse $(i_{0}+1)$-gon
$\mathsf{X}_{1}^{-1}*\mathsf{Y}_{1}*\dots\mathsf{Y}_{i_{0}-1}*\mathsf{Z}_{1}*$
and we conclude by induction that every $\mathsf{Y}_{t_{j}}$ with
$t_{j}<i_{0}$ has a vertex close to a vertex $\mathsf{a}_{j}$ on $\mathsf{X}$
and the vertices $\mathsf{a}_{j}$ occur in $\mathsf{X}$ in the appropriate
order. Similarly, we conclude the same for every path $\mathsf{Y}_{t_{j}}$
with $t_{j}>i_{0}$. This implies the statement for all $\mathsf{Y}_{t_{j}}$. ∎

######  11.3 Lemma.

Let $X$ be a word reduced in $G_{\alpha-1}$. Assume that for any fragment $K$
of rank $\alpha$ in $X$ we have

$\mu_{\mathrm{f}}(K)\leq 1-3\lambda-5\omega.$

Then there exists a word $Y$ equal to $X$ in $G_{\alpha}$ which is reduced in
$G_{\alpha-1}$ and such that for any fragment $M$ of rank $\alpha$ in $Y$ we
have

$\mu_{\mathrm{f}}(M)<\frac{1}{2}+2\lambda+15\omega.$

In particular, $Y$ is reduced in $G_{\alpha}$ (note that
$\frac{1}{2}+2\lambda+15\omega<\rho=1-9\lambda$ by (2-3) and (4-1).)

###### Proof.

We represent $X$ by a reduced path $\mathsf{X}$ in $\Gamma_{\alpha-1}$. Denote

$t=\frac{1}{2}+11\omega.$

Let $\mathsf{K}_{1}$, $\dots$, $\mathsf{K}_{r}$ be a maximal set of pairwise
non-compatible fragments of rank $\alpha$ in $\mathsf{X}$ with
$\mu_{\mathrm{f}}(\mathsf{K}_{i})\geq t$. We assume that each $\mathsf{K}_{i}$
has maximal size $\mu_{\mathrm{f}}(\mathsf{K}_{i})$ in its equivalence class
of compatible fragments of rank $\alpha$ occurring in $\mathsf{X}$. Using
Proposition 8.12 we shorten each $\mathsf{K}_{i}$ from the start obtaining a
fragment $\bar{\mathsf{K}}_{i}$ of rank $\alpha$ so that
$\bar{\mathsf{K}}_{i}$ do not intersect pairwise; we have
$\mu_{\mathrm{f}}(\bar{\mathsf{K}}_{i})>\mu_{\mathrm{f}}(\mathsf{K}_{i})-\lambda-2.7\omega$.
Let

$\mathsf{X}=\mathsf{S}_{0}\bar{\mathsf{K}}_{1}\mathsf{S}_{1}\dots\bar{\mathsf{K}}_{r}\mathsf{S}_{r}.$

Let $\mathsf{P}_{i}$ be a base for $\bar{\mathsf{K}}_{i}$; for each $i$, we
have a coarse bigon
$\bar{\mathsf{K}}_{i}^{-1}\mathsf{u}_{i}\mathsf{P}_{i}\mathsf{v}_{i}$ with
bridges $\mathsf{u}_{i}$ and $\mathsf{v}_{i}$. Let
$P_{i}\eqcirc\operatorname{\mathit{label}}(\mathsf{P}_{i})$ and
$P_{i}Q_{i}^{-1}$ be the associated relator of rank $\alpha$. We consider a
path in $\Gamma_{\alpha-1}$

$\mathsf{Z}=\mathsf{S}_{0}^{*}\mathsf{u}_{1}^{*}\mathsf{Q}_{1}\mathsf{v}_{1}^{*}\mathsf{S}_{1}^{*}\dots\mathsf{u}_{r}^{*}\mathsf{Q}_{r}\mathsf{v}_{r}^{*}\mathsf{S}_{r}^{*}$

where labels of $\mathsf{S}_{i}^{*}$, $\mathsf{u}_{i}^{*}$ and
$\mathsf{v}_{i}^{*}$ are equal to corresponding labels of $\mathsf{S}_{i}$,
$\mathsf{u}_{i}$ and $\mathsf{v}_{i}$ and
$\operatorname{\mathit{label}}(\mathsf{Q}_{i})\eqcirc Q_{i}$. Note that
$\operatorname{\mathit{label}}(\mathsf{Z})=X$ in $G_{\alpha}$. We perform the
following procedure:

  1. (i)

if a pair of vertices on $\mathsf{Q}_{i}$ and $\mathsf{S}_{i}^{*}$ are close
and is distinct from $(\tau(\mathsf{Q}_{i}),\iota(\mathsf{S}_{i}^{*}))$ then
we choose a bridge $\mathsf{w}$ of rank $\alpha-1$ joining these vertices,
replace $\mathsf{v}_{i}^{*}$ with $\mathsf{w}$ and shorten $\mathsf{Q}_{i}$
from the end and $\mathsf{S}_{i}^{*}$ from the start; similarly, if a pair of
vertices on $\mathsf{Q}_{i}$ and $\mathsf{S}_{i-1}^{*}$ are close and is
distinct from $(\iota(\mathsf{Q}_{i}),\tau(\mathsf{S}_{i-1}^{*}))$ then we
choose a bridge $\mathsf{w}$ of rank $\alpha-1$ joining them and replace
$\mathsf{u}_{i}^{*}$ with $\mathsf{w}$ shortening $\mathsf{Q}_{i}$ from the
start and $\mathsf{S}_{i-1}^{*}$ from the end; we apply recursively the
operation until possible;

  2. (ii)

if a vertex on $\mathsf{Q}_{i}$ is close to a vertex on $\mathsf{Q}_{i+1}^{*}$
then we choose a bridge $\mathsf{w}$ of rank $\alpha-1$ joining these
vertices, shorten $\mathsf{Q}_{i}$ from the end and $\mathsf{Q}_{i+1}$ from
the end and join then by $\mathsf{w}$ (so $\mathsf{S}_{i}^{*}$ is eliminated
and $\mathsf{v}_{i}^{*}\mathsf{S}_{i}^{*}\mathsf{u}_{i}^{*}$ is replaced with
a bridge $\mathsf{w}$ of rank $\alpha-1$); we apply recursively the operation
until possible;

After the procedure, we obtain a path

$\mathsf{Z}_{1}=\mathsf{T}_{0}\mathsf{U}_{0}\mathsf{R}_{1}\mathsf{U}_{1}\dots\mathsf{R}_{r}\mathsf{U}_{r}\mathsf{T}_{r}$

where for each $i$, $\mathsf{R}_{i}$ is a subpath of $\mathsf{Q}_{i}$ and
$\mathsf{U}_{i}$ either is a bridge of rank $\alpha-1$ or has the form
$\mathsf{w}_{i}\mathsf{T}_{i}\mathsf{z}_{i}$ where $\mathsf{T}_{i}$ is a
subpath of $\mathsf{S}_{i}^{*}$ and $\mathsf{w}_{i}$ and $\mathsf{z}_{i}$ are
bridges of rank $\alpha-1$. Let $\mathsf{Y}$ be a reduced path with the same
endpoints as $\mathsf{Z}_{1}$. Our goal is to prove that the label $Y$ of
$\mathsf{Y}$ satisfies the requirement of the lemma, that is, for any fragment
$\mathsf{N}$ of rank $\alpha$ in $\mathsf{Y}$ we have
$\mu_{\mathrm{f}}(\mathsf{N})<\frac{1}{2}+2\lambda+15\omega$.

We compute a lower bound for $\mu(\mathsf{R}_{i})$. Fix $i$ and let
$\mathsf{Q}_{i}=\mathsf{Q}^{\prime}\mathsf{R}_{i}\mathsf{Q}^{\prime\prime}$.
At step (i) of the procedure, we do not shorten $\mathsf{Q}_{i}$ more than
this would give a fragment of rank $\alpha$ in $\mathsf{X}$ with a base that
is a proper extension of $\mathsf{P}_{i}$, so we get $\mu(\mathsf{Q}_{i})\geq
1-\mu_{\mathrm{f}}(\mathsf{K}_{i})\geq 3\lambda+5\omega$. At step (ii) we
shorten $\mathsf{Q}_{i}$ from each side by less than $\lambda+0.4\omega$ (this
follows from Proposition 9.19(i)α-1, Proposition 8.15 and Corollary 8.2). This
implies $\mu(\mathsf{R}_{i})>\lambda+4\omega$ and, in particular,
$|\mathsf{R}_{i}|_{\alpha-1}>4\eta$.

We apply Lemma 11.2 with $\mathsf{X}:=\mathsf{Y}$ where $\mathsf{R}_{i}$ and
$\mathsf{T}_{i}$ play the role of $\mathsf{Y}_{i}$’s and $\mathsf{R}_{i}$ are
taken as $\mathsf{Y}_{t_{i}}$. The lemma says that each path $\mathsf{R}_{i}$
has a vertex close to a vertex on $\mathsf{Y}$ and these vertices on
$\mathsf{Y}$ are appropriately ordered. We can write

$\mathsf{Y}=\mathsf{V}_{0}\mathsf{M}_{1}\mathsf{V}_{1}\dots\mathsf{M}_{r}\mathsf{V}_{r}$

where each $\mathsf{M}_{i}$ is close to a subpath of $\mathsf{Q}_{i}$ (at the
moment each $\mathsf{M}_{i}$ is empty because it is represented by a vertex on
$\mathsf{Y}$). Extending $\mathsf{M}_{i}$’s we make them maximal so that no
vertex on $\mathsf{W}_{i}$ except $\iota(\mathsf{V}_{i})$ is close to a vertex
on $\mathsf{Q}_{i}$ and no vertex on $\mathsf{V}_{i}$ except
$\tau(\mathsf{V}_{i})$ is close to a vertex on $\mathsf{Q}_{i+1}$. Up to
location of $\mathsf{Z}$ in $\Gamma_{\alpha-1}$ we can assume that it starts
at $\iota(\mathsf{X})$. Combining the two graphs shown in Figure 33a and
mapping them to $\Gamma_{\alpha}$ we obtain a graph as shown in Figure 33b.

$\mathsf{u}_{1}$
---
$\mathsf{v}_{1}$
---
$\mathsf{u}_{2}$
---
$\mathsf{v}_{2}$
---
$\mathsf{u}_{r}$
---
$\mathsf{v}_{r}$
---
$\mathsf{S}_{0}$
---
$\mathsf{S}_{1}$
---
$\mathsf{S}_{2}$
---
$\mathsf{S}_{r-1}^{*}$
---
$\mathsf{S}_{r}$
---
$\mathsf{u}_{1}^{*}$
---
$\mathsf{v}_{1}^{*}$
---
$\mathsf{u}_{2}^{*}$
---
$\mathsf{v}_{2}^{*}$
---
$\mathsf{u}_{r}^{*}$
---
$\mathsf{v}_{r}^{*}$
---
$\mathsf{S}_{0}^{*}$
---
$\mathsf{S}_{1}^{*}$
---
$\mathsf{S}_{2}^{*}$
---
$\mathsf{S}_{r-1}^{*}$
---
$\mathsf{S}_{r}^{*}$
---
$\mathsf{Q}_{1}$
---
$\mathsf{Q}_{2}$
---
$\mathsf{Y}$
---
$\mathsf{T}_{0}$
---
$\mathsf{U}_{0}$
---
$\mathsf{R}_{1}$
---
$\mathsf{U}_{1}$
---
$\mathsf{R}_{2}$
---
$\mathsf{U}_{2}$
---
$\mathsf{U}_{r-1}$
---
$\mathsf{R}_{r}$
---
$\mathsf{U}_{r}$
---
$\mathsf{T}_{r}$
---
a
---
b
---
Figure 33.

This graph is similar to one obtained from a single-layer diagram (as in
Figure 15). An easy analysis with use of Proposition 9.19α-1, Proposition 8.15
and Corollary 8.2 shows that $\mathsf{M}_{i}$ and some extension
$\tilde{\mathsf{K}}_{i}$ of $\bar{\mathsf{K}}_{i}$ satisfy the bound as in
Proposition 9.7, i.e.

$\mu_{\mathrm{f}}(\mathsf{M}_{i})+\mu_{\mathrm{f}}(\tilde{\mathsf{K}}_{i})>1-2\lambda-1.5\omega.$

Since
$\mu_{\mathrm{f}}(\tilde{\mathsf{K}}_{i})\leq\mu_{\mathrm{f}}(\mathsf{K}_{i})\leq
1-3\lambda-5\omega$ we obtain that for all $i$,

$\mu_{\mathrm{f}}(\mathsf{M}_{i})>\lambda+3.5\omega.$

Let $\mathsf{N}$ be a fragment of rank $\alpha$ in $\mathsf{Y}$. By
Proposition 8.10, we have either $\mathsf{N}\sim\mathsf{M}_{i}$ or
$\mathsf{N}\subseteq\mathsf{M}_{i}\cup\mathsf{M}_{i+1}$ for some $i$. In the
case when $\mathsf{N}\subseteq\mathsf{M}_{i}\cup\mathsf{M}_{i+1}$,
$\mathsf{N}\not\sim\mathsf{M}_{i}$ and $\mathsf{N}\not\sim\mathsf{M}_{i+1}$ we
can apply the argument from the proof of Proposition 10.5 and find a fragment
$\mathsf{N}^{\prime}$ in $\mathsf{X}$ such that

$\mu_{\mathrm{f}}(\mathsf{N}^{\prime})>\mu_{\mathrm{f}}(\mathsf{N})-2\lambda-3.4\omega.$

We have also $\mathsf{N}^{\prime}\not\sim\mathsf{K}_{i},\mathsf{K}_{i+1}$ and
hence $\mathsf{N}^{\prime}\not\sim\mathsf{K}_{j}$ for all $j$. By the choice
of the $\mathsf{K}_{i}$’s, we have $\mu_{\mathrm{f}}(\mathsf{K}^{\prime})<t$
and hence

$\mu_{\mathrm{f}}(\mathsf{N})<t+2\lambda+3.4\omega<\frac{1}{2}+2\lambda+15\omega.$

Assume that $\mathsf{N}\sim\mathsf{M}_{i}$ for some $i$. Let
$\bar{\mathsf{Q}}$ and $\bar{\mathsf{P}}$ be bases for $\mathsf{N}$ and
$\mathsf{K}_{i}$ respectively. Images of $\bar{\mathsf{Q}}^{-1}$ and
$\bar{\mathsf{P}}$ in $\Gamma_{\alpha}$ are subpaths of a relator loop and
have at most two overlapping parts. We give an upper bound for
$\mu(\bar{\mathsf{Q}})+\mu(\bar{\mathsf{P}})$ by finding an upper bound for
the size of each overlapping part. Assume, for example, that an end of the
image of $\bar{\mathsf{P}}$ in $\Gamma_{\alpha}$ overlaps with a start of the
image of $\bar{\mathsf{Q}}^{-1}$. Changing the location of $\mathsf{Z}$ in
$\Gamma_{\alpha-1}$ we can assume that $\bar{\mathsf{P}}$ and
$\bar{\mathsf{Q}}^{-1}$ overlap on a subpath $\mathsf{W}$ of the same size
already in $\Gamma_{\alpha-1}$.

We consider the case $i<r$ (see Figure 34; the case $i=r$ is similar with a
better upper bound on $\mu(\mathsf{W})$).

$\mathsf{K}_{i}$
---
$\bar{\mathsf{K}}_{i+1}$
---
$\mathsf{S}$
---
$\mathsf{X}$
---
$\bar{\mathsf{P}}$
---
$\bar{\mathsf{Q}}$
---
$\mathsf{W}$
---
$\mathsf{V}$
---
$\mathsf{N}$
---
$\mathsf{L}$
---
$\mathsf{Y}$
---
$\mathsf{P}_{i+1}$
---
$\mathsf{M}_{i+1}$
---
Figure 34.

We apply Proposition 9.19(ii)α-1 to a coarse tetragon with one side
$\mathsf{W}$ and other sides which are an end $\mathsf{S}$ of
$\mathsf{S}_{i}\bar{\mathsf{K}}_{i+1}$, a start $\mathsf{V}$ of
$\mathsf{M}_{i+1}^{-1}\mathsf{V}_{i}^{-1}$ and a subpath of a common base axis
$\mathsf{L}$ for $\mathsf{K}_{i+1}^{-1}$ and $\mathsf{N}_{i+1}$. In the worst
case we have
$\mathsf{W}=\mathsf{W}_{1}\mathsf{z}_{1}\mathsf{W}_{2}\mathsf{z}_{2}\mathsf{W}_{3}$
where $\mathsf{W}_{1}$ is close to a subpath of $\mathsf{V}^{-1}$,
$\mathsf{W}_{2}$ is close to a subpath of $\mathsf{L}^{-1}$, $\mathsf{W}_{3}$
is close to a subpath of $\mathsf{S}^{-1}$ and
$|\mathsf{z}_{i}|_{\alpha-1}\leq 4\eta\zeta$. Proposition 10.21α-1 implies
$|\mathsf{W}_{1}|_{\alpha-1}<5.7$ and $|\mathsf{W}_{3}|_{\alpha-1}<5.7$. Since
$\mathsf{K}_{i}\not\sim\mathsf{K}_{i+1}$ we obtain
$\mu(\mathsf{W}_{2})<\lambda$. Hence

$\mu(\mathsf{W})<\lambda+2\omega(5.7+4\eta\zeta)<\lambda+13\omega.$

We obtain

$\mu_{\mathrm{f}}(\mathsf{N})+\mu_{\mathrm{f}}(\mathsf{K}_{i})<1+2\lambda+26\omega.$

Since $\mu_{\mathrm{f}}(\mathsf{K}_{i})\geq t$ this implies the required bound
$\mu_{\mathrm{f}}(\mathsf{N})<\frac{1}{2}+2\lambda+15\omega$. ∎

######  11.4 Lemma.

Let $\alpha\geq 1$ and $X$ be a word reduced in $G_{\alpha}$ and
$a\in\mathcal{A}^{\pm 1}$ a letter in the generators of $G_{\alpha}$. Let $Y$
be a word reduced in $G_{\alpha-1}$ such that $Y=Xa$ in $G_{\alpha-1}$. Then
$Y$ has no fragments $K$ of rank $\alpha$ with
$\mu_{\mathrm{f}}(K)\geq\rho+6.2\omega$.

###### Proof.

Follows from Lemma 10.8 and Proposition 8.8. ∎

###### Proof of Proposition 11.1.

It is trivial if $\alpha=0$. In the case $\alpha\geq 1$ Proposition 11.1
follows by induction from Lemmas 11.3 and 11.4 since
$\rho+6.2\omega<1-3\lambda-5\omega.$ ∎

We turn to the cyclic analogue of Proposition 11.1:

######  11.5 Proposition (cyclically reduced representative).

Every element of $G_{\alpha}$ of finite order is conjugate to a cyclically
reduced word of the form $R_{0}^{k}$ where $R_{0}$ is the root of a relator of
rank $\beta$, $1\leq\beta\leq\alpha$.

Every element of $G_{\alpha}$ of infinite order is conjugate to a strongly
cyclically reduced word in $G_{\alpha}$.

######  11.6 Lemma (a cyclic version of Lemma 11.2).

Let $X$ be a word cyclically reduced in $G_{\alpha-1}$ representing an element
of $G_{\alpha-1}$ of infinite order. Let $m\geq 2$, $Y_{1},\dots,Y_{m}$ be
words reduced in $G_{\alpha-1}$, $u_{1},\dots,u_{m}$ be bridges of rank
$\alpha-1$ and let $X$ be conjugate to $Y_{1}u_{1}\dots Y_{m}u_{m}$ in
$G_{\alpha-1}$. Let
$\prod_{i\in\mathbb{Z}}\mathsf{Y}_{1}^{(i)}\mathsf{u}_{1}^{(i)}\dots\mathsf{Y}_{m}^{(i)}\mathsf{u}_{m}^{(i)}$
and $\bar{\mathsf{X}}=\prod_{i\in\mathbb{Z}}\mathsf{X}^{(i)}$ be lines in
$\Gamma_{\alpha-1}$ labeled $(Y_{1}u_{1}\dots Y_{m}u_{m})^{\infty}$ and
$X^{\infty}$ respectively representing the conjugacy relation.

Assume that there are indices $1\leq t_{1}<t_{2}<\dots<t_{k}\leq m$ $(k\geq
1)$ such that

$m+t_{1}-t_{m}\leq 2,\quad t_{j}-t_{j-1}\leq 2\quad\text{ for all }j,$

and

$|Y_{t_{j}}|_{\alpha-1}>4\eta\quad\text{for all }j.$

Assume that there are no close vertices in each of the pairs
$(\mathsf{Y}_{i}^{(0)},\mathsf{Y}_{i+1}^{(0)})$,
$(\mathsf{Y}_{m}^{(0)},\mathsf{Y}_{1}^{(1)})$,
$(\mathsf{Y}_{t_{j}}^{(0)},\mathsf{Y}_{t_{j}+1}^{(0)})$,
$(\mathsf{Y}_{t_{k}}^{(0)},\mathsf{Y}_{t_{1}}^{(1)})$ except appropriate
endpoints (i.e. except pairs
$(\tau(\mathsf{Y}_{i}^{(0)}),\iota(\mathsf{Y}_{i+1}^{(0)}))$ and
$(\tau(\mathsf{Y}_{m}^{(0)}),\iota(\mathsf{Y}_{1}^{(1)}))$). Then each of the
paths $\mathsf{Y}_{t_{j}}^{(0)}$, $j=1,\dots,k$ has a vertex close to a vertex
$\mathsf{a}_{j}$ on $\bar{\mathsf{X}}$ and these vertices $\mathsf{a}_{j}$ are
in the (non-strict) order corresponding to the order of the
$\mathsf{Y}_{j}^{(0)}$’s (and $\mathsf{a}_{k}$ is located non-strictly before
$s_{X,\bar{\mathsf{X}}}\mathsf{a}_{0}$).

###### Proof.

The proof follows the proof of Lemma 11.2 with appropriate changes.

  * Claim 1:

There are no close vertices in pairs
$(\mathsf{Y}_{i}^{(0)},\mathsf{Y}_{j}^{(0)})$ with $j-i>1$ and
$(\mathsf{Y}_{i}^{(0)},\mathsf{Y}_{j}^{(1)})$ with $j+m-i>1$.

The proof repeats the argument from the proof of Lemma 11.2.

  * Claim 2:

For some $i$, there are close vertices in the pair
$(\mathsf{Y}_{i}^{(0)},\bar{\mathsf{X}})$.

Assume this is not true. Consider an annular diagram $\Delta$ of rank
${\alpha-1}$ with boundary loops $\hat{\mathsf{X}}^{-1}$ and
$\hat{\mathsf{Y}}_{1}\hat{\mathsf{u}}_{1}\dots\hat{\mathsf{Y}}_{m}\hat{\mathsf{u}}_{m}$
and a combinatorially continuous map $\phi:\tilde{\Delta}\to\Gamma_{\alpha-1}$
such that $\phi$ maps the boundary of $\tilde{\Delta}$ to
$\bar{\mathsf{X}}^{-1}$ and
$\prod_{i}\mathsf{Y}_{1}^{(i)}\mathsf{u}_{1}^{(i)}\dots\mathsf{Y}_{m}^{(i)}\mathsf{u}_{m}^{(i)}$.
The assumption, Claim 1 and the hypothesis of the lemma imply that $\Delta$ is
small. Application of Proposition 7.9α-1 gives

$\sum_{i}|Y_{i}|_{\alpha-1}\leq\eta m.$

On the other hand, from the hypothesis of the lemma we have
$\sum_{i}|Y_{i}|_{\alpha-1}\geq 4k\eta>\eta m$, a contradiction. This proves
the claim.

By Claim 2, assume without loss of generality that there is a vertex
$\mathsf{b}$ on $\mathsf{Y}_{1}^{(0)}$ which is close to a vertex $\mathsf{c}$
on $\bar{\mathsf{X}}$. Let $\mathsf{b}$ divide $\mathsf{Y}_{1}^{(0)}$ as
$\mathsf{Y}_{1}^{(0)}=\mathsf{Z}_{1}\mathsf{Z}_{2}$ and up to cyclic shift of
$X$, assume that $\mathsf{X}^{(0)}$ starts at $\mathsf{c}$. Now we can
directly apply Lemma 11.2 to the coarse $(m+2)$-gon

$(\mathsf{X}^{(0)})^{-1}*\mathsf{Z}_{2}\mathsf{u}_{1}^{(0)}\mathsf{Y}_{2}^{(0)}\dots\mathsf{u}_{m-1}^{(0)}\mathsf{Y}_{m}^{(0)}\mathsf{u}_{m}^{(0)}\mathsf{Z}_{1}*$

and get the required conclusion. ∎

######  11.7 Lemma (a cyclic version of Lemma 11.3).

Let $X$ be a word strongly cyclically reduced in $G_{\alpha-1}$. Assume that
$X$ is not conjugate in $G_{\alpha}$ to a power of the root of a relator of
rank $\beta\leq\alpha$. Next, assume that for any fragment $K$ of rank
$\alpha$ in a cyclic shift of $X$ we have

$\mu_{\mathrm{f}}(K)\leq 1-4\lambda-8\omega.$

Then there exists a word $Z$ conjugate to $X$ in $G_{\alpha}$ which is
strongly cyclically reduced in $G_{\alpha-1}$ and such that no power $Z^{k}$
contains a fragment $L$ of rank $\alpha$ with

$\mu_{\mathrm{f}}(L)<\frac{1}{2}+2\lambda+15\omega.$

In particular, $Z$ is strongly cyclically reduced in $G_{\alpha}$.

###### Proof.

The general scheme is the same as in the proof of Lemma 11.3. Let
$\bar{\mathsf{X}}=\prod_{i\in\mathbb{Z}}\mathsf{X}_{i}$ be a line in
$\Gamma_{\alpha-1}$ labeled $X^{\infty}$. First we note that for any fragment
$\mathsf{K}$ of rank $\alpha$ in $\bar{\mathsf{X}}$ we have
$s_{X,\bar{\mathsf{X}}}\mathsf{K}\not\sim\mathsf{K}$ by Proposition 8.16(ii).
By Propositions 8.10 and 8.11 there exists a starting segment
$\mathsf{K}^{\prime}$ of $\mathsf{K}$ that is a fragment of rank $\alpha$ with
$\mu_{\mathrm{f}}(\mathsf{K}^{\prime})>\mu_{\mathrm{f}}(\mathsf{K})-\lambda-3\omega$
and $|\mathsf{K}^{\prime}|\leq|X|$, i.e.
$\operatorname{\mathit{label}}(\mathsf{K}^{\prime})$ occurs in a cyclic shift
of $X$. Then the hypothesis of the lemma implies that $\bar{\mathsf{X}}$
contains no fragments $\mathsf{K}$ of rank $\alpha$ with
$\mu_{\mathrm{f}}(\mathsf{K})\geq 1-3\lambda-5\omega$.

Denote $t=\frac{1}{2}+11\omega$. We can assume that there is at least one
fragment $\mathsf{K}$ of rank $\alpha$ in $\bar{\mathsf{X}}$ with
$\mu_{\mathrm{f}}(\mathsf{K})\geq t$ (otherwise we can take $Z:=X$). We choose
a maximal set $\mathsf{K}_{1}$, $\dots$, $\mathsf{K}_{r}$ of pairwise non-
compatible fragments of rank $\alpha$ in $\bar{\mathsf{X}}$ with
$\mu_{\mathrm{f}}(\mathsf{K}_{i})\geq t$ such that
$\mathsf{K}_{1}<\dots<\mathsf{K}_{r}<s_{X,\bar{\mathsf{X}}}\mathsf{K}_{1}$ and
$\mathsf{K}_{r}\not\sim s_{X,\bar{\mathsf{X}}}\mathsf{K}_{1}$ (after choosing
$\mathsf{K}_{1}$ we use Proposition 8.16(ii) to get
$s_{X,\bar{\mathsf{X}}}\mathsf{K}_{1}\not\sim\mathsf{K}_{1}$). We assume that
each $\mathsf{K}_{i}$ has maximal size $\mu_{\mathrm{f}}(\mathsf{K}_{i})$ in
its class of compatible fragments of rank $\alpha$ in $\bar{\mathsf{X}}$.
Using Proposition 8.12 we shorten each $\mathsf{K}_{i}$ from its start
obtaining a fragment $\bar{\mathsf{K}}_{i}$ of rank $\alpha$ so that all
$\bar{\mathsf{K}}_{i}$ do not intersect pairwise and
$|\mathsf{K}_{1}\cup\mathsf{K}_{r}|\leq|X|$; we have
$\mu_{\mathrm{f}}(\bar{\mathsf{K}}_{i})>\mu_{\mathrm{f}}(\mathsf{K}_{i})-\lambda-2.7\omega$.
Passing to a cyclic shift of $X$ (and changing all $\mathsf{X}_{i}$
accordingly) we may assume also that

$\mathsf{X}_{0}=\bar{\mathsf{K}}_{1}\mathsf{S}_{1}\dots\bar{\mathsf{K}}_{r}\mathsf{S}_{r}.$

Let $\mathsf{P}_{i}$ be the base for $\bar{\mathsf{K}}_{i}$ and
$\bar{\mathsf{K}}_{i}^{-1}\mathsf{u}_{i}\mathsf{P}_{i}\mathsf{v}_{i}$ a loop
in $\Gamma_{\alpha-1}$ with bridges $\mathsf{u}_{i}$ and $\mathsf{v}_{i}$.
Denote $S_{i}\eqcirc\operatorname{\mathit{label}}(\mathsf{S}_{i})$,
$P_{i}\eqcirc\operatorname{\mathit{label}}(\mathsf{P}_{i})$,
$u_{i}\eqcirc\operatorname{\mathit{label}}(\mathsf{u}_{i})$,
$v_{i}\eqcirc\operatorname{\mathit{label}}(\mathsf{v}_{i})$ and let
$P_{i}Q_{i}^{-1}$ be the associated relator of rank $\alpha$. Let

$Z=u_{1}Q_{1}v_{1}S_{1}u_{2}Q_{1}v_{2}S_{2}\dots u_{r}Q_{r}v_{r}S_{r}.$

Let $Y$ be a word strongly cyclically reduced in $G_{\alpha-1}$ that is
conjugate to $Z$ in $G_{\alpha-1}$. We prove that $Y$ satisfies the
requirements of the lemma. Note that $Y$ and hence $Z$ are conjugate to $X$ in
$G_{\alpha}$.

We transform $Z$ using a procedure analogous to the procedure described in the
proof of Lemma 11.3. At any moment, we will have a word $Z_{1}$ of the form

$Z_{1}=R_{1}U_{1}\dots R_{r}U_{r},$

conjugate to $Z$ in $G_{\alpha-1}$ where each $R_{i}$ is a subword of $Q_{i}$
and each $U_{i}$ either is a bridge of rank $\alpha-1$ or has the form
$w_{i}T_{i}z_{i}$ where $w_{i}$, $z_{i}$ are bridges of rank $\alpha-1$ and
$T_{i}$ is a subword of $S_{i}$. At the start, we have $R_{i}=Q_{i}$ and
$U_{i}=v_{i}S_{i}u_{i+1}$ (here and below $i+1$ is taken modulo $r$). The
transformation procedure consists of the following steps applied recursively
until possible.

  1. (i)

Suppose that $U_{i}$ has the form $w_{i}T_{i}z_{i}$ above. If
$R_{i}=R^{\prime}R^{\prime\prime}$, $T_{i}=T^{\prime}T^{\prime\prime}$ where
$|R^{\prime\prime}|+|T^{\prime}|>0$ and $R^{\prime\prime}w_{i}T^{\prime}$ is
equal in $G_{\alpha-1}$ to a bridge $w$ of rank $\alpha-1$ then replace
$R_{i}$, $w_{i}$ and $T_{i}$ with $R^{\prime}$, $w$ and $T^{\prime\prime}$
respectively; similarly, if $T_{i}=T^{\prime}T^{\prime\prime}$,
$R_{i+1}=R^{\prime}R^{\prime\prime}$ where $|T^{\prime\prime}|+|R^{\prime}|>0$
and $T^{\prime\prime}z_{i}R^{\prime}$ is equal in $G_{\alpha-1}$ to a bridge
$w$ of rank $\alpha-1$ then replace $T_{i}$, $z_{i}$ and $R_{i+1}$ with
$T^{\prime}$, $w$ and $R^{\prime\prime}$ respectively.

  2. (ii)

If $R_{i}=R^{\prime}R^{\prime\prime}$ and $R_{i+1}=R^{*}R^{**}$ where
$|R^{\prime\prime}|+|R^{*}|>0$ and $R^{\prime\prime}U_{i}R^{*}$ is equal in
$G_{\alpha-1}$ to a bridge $w$ of rank $\alpha-1$ then replace $R_{i}$,
$U_{i}$ and $R_{i+1}$ with $R^{\prime}$, $w$ and $R^{**}$ respectively.

Similar to the proof of Lemma 11.3, after performing the procedure we obtain
$|R_{i}|_{\alpha-1}>4\eta$ for all $i$.

Let $\bar{\mathsf{Z}}=\prod_{i\in\mathbb{Z}}\mathsf{Z}^{(i)}$ be a line in
$G_{\alpha-1}$ labeled $Z^{\infty}$ and let $\mathsf{Q}_{j}^{(i)}$ denote the
appropriate subpath of $\mathsf{Z}^{(i)}$ labeled $Q_{j}$. We can implement
the procedure above on the line $\bar{\mathsf{Z}}$ instead of a word $Z$ by
changing appropriate paths instead of words (to each change of words in (i) or
(ii) there corresponds infinitely many changes of paths translated by
$s_{X,\bar{\mathsf{X}}}$). As a result, we get a line
$\prod_{i\in\mathbb{Z}}\mathsf{Z}_{1}^{(i)}$ so that the corresponding subpath
$\mathsf{R}_{j}^{(i)}$ of $\mathsf{Z}_{1}^{(i)}$ is also a subpath of
$\mathsf{Q}_{j}^{(i)}$. Denote also $\mathsf{T}_{j}^{(i)}$ the appropriate
subpath of $\mathsf{Z}_{1}^{(i)}$ labeled $T_{j}$. Let
$\bar{\mathsf{Y}}=\prod_{i\in\mathbb{Z}}\mathsf{Y}^{(i)}$ be the line in
$G_{\alpha-1}$ such that $\bar{\mathsf{Z}}$ and $\bar{\mathsf{Y}}$ are
associated with conjugate words $Z$ and $Y$. We apply Lemma 11.6 with
$\bar{\mathsf{X}}:=\bar{\mathsf{Y}}$ where $\mathsf{R}_{j}^{(i)}$ and
$\mathsf{T}_{j}^{(i)}$ play the role of $\mathsf{Y}_{j}^{(i)}$’s and
$\mathsf{R}_{j}^{(i)}$ are taken as $\mathsf{Y}_{t_{j}}^{(i)}$. According to
the lemma, each path $\mathsf{R}_{j}^{(0)}$ has a vertex close to a vertex on
$\bar{\mathsf{Y}}$, these vertices on $\bar{\mathsf{Y}}$ are ordered along
$\bar{\mathsf{Y}}$ in the increasing order of the index $j$, and the length of
the segment of $\bar{\mathsf{Y}}$ between the first and the last one is not
more that $|Y|$. Up to cyclic shift of $Y$, we can write

$\mathsf{Y}^{(0)}=\mathsf{W}_{0}\mathsf{M}_{1}\mathsf{W}_{1}\dots\mathsf{M}_{r}\mathsf{W}_{r}$

where each $\mathsf{M}_{j}$ is close to a subpath of $\mathsf{Q}_{j}^{(0)}$.
Taking $\mathsf{M}_{j}$ maximal with these properties we obtain, as in the
proof of Lemma 11.3,

$\mu_{\mathrm{f}}(\mathsf{M}_{i})>\lambda+3.5\omega\quad\text{for all }j.$

The rest of the proof is similar to the proof of Lemma 11.3. ∎

######  11.8 Lemma.

If $\mathsf{X}$ is a reduced path in $\Gamma_{\alpha}$ and the endpoints of
$\mathsf{X}$ are close then $|\mathsf{X}|_{\alpha}\leq 1$.

###### Proof.

For $\alpha\geq 1$ this follows from Lemma 9.22. ∎

######  11.9 Lemma.

If $P$ is a piece of rank $\alpha$ then for any fragment $K$ of rank $\alpha$
in $P$ we have $\mu_{\mathrm{f}}(K)\leq\max\\{\lambda,\mu(P)+2\omega\\}$.

###### Proof.

Let $\mathsf{P}$ be a path in $\Gamma_{\alpha-1}$ with
$\operatorname{\mathit{label}}(\mathsf{P})\eqcirc P$, let $R$ be the
associated relator of rank $\alpha$ and let $\mathsf{L}$ be the line labeled
$R^{\infty}$ extending $\mathsf{P}$. Assume that $\mathsf{K}$ is a fragment of
rank $\alpha$ contained in $\mathsf{P}$. If the base axis for $\mathsf{K}$ is
distinct from $\mathsf{L}$ then $\mu_{\mathrm{f}}(\mathsf{K})<\lambda$ by
Corollary 8.2. Otherwise the base $\mathsf{Q}$ for $\mathsf{K}$ is contained
in $\mathsf{L}$ and Lemma 11.8α-1 implies

$\mu_{\mathrm{f}}(\mathsf{K})=\mu(\mathsf{Q})\leq\mu(\mathsf{K})+2\omega\leq\mu(\mathsf{P})+2\omega.$

∎

######  11.10 Proposition.

Let $P$ be a piece of rank $1\leq\beta\leq\alpha$ with
$\mu(P)\leq\rho-2\omega$. Then $P$ is reduced in $G_{\alpha}$. If $R\eqcirc
QS$ where $R$ is a relator of rank $\beta$ then either $Q$ or $S$ is reduced
in $G_{\alpha}$.

###### Proof.

The first statement follows from Lemmas 10.8 and 11.9. If $R$ is a relator of
rank $\beta$ and $R\eqcirc QS$ then by 4.14(ii), we have either
$\mu(Q)\leq\frac{1}{2}+\omega$ or $\mu(S)\leq\frac{1}{2}+\omega$. It remains
to note that $\frac{1}{2}+\omega<\rho-2\omega$. ∎

###### Proof of Proposition 11.5.

Let $X$ be a word representing an element of $G_{\alpha}$. We may assume that
$X$ is reduced in $G_{\alpha}$ as a non-cyclic word. We perform a ‘‘coarse
cyclic cancellation’’ in $X$: represent $X$ as $UX_{1}V$ where $VU$ is equal
in $G_{\alpha}$ to a bridge $u$ of rank $\alpha$ and $X_{1}$ has the minimal
possible length. Let $u\eqcirc v_{1}Pv_{2}$ where $P$ is a piece of rank
$\alpha$. We can assume that $\mu(P)\leq\frac{1}{2}+\omega$. Let $Y$ be a word
cyclically reduced in $G_{\alpha-1}$ and conjugate to $X_{1}u$ in
$G_{\alpha-1}$. Note that $X_{1}u$ and hence $Y$ are conjugate to $X$ in
$G_{\alpha}$. We show that either $Y$ is conjugate in $G_{\alpha-1}$ to a
power $R_{0}^{t}$ of the root $R_{0}$ of a relator of rank $\beta\leq\alpha$
or no cyclic shift of $Y$ contains a fragment $K$ of rank $\alpha$ with
$\mu_{\mathrm{f}}(\mathsf{K})\geq\rho+2\lambda+16\omega$. In the first case,
by Proposition 11.10 we can assume that $R_{0}^{k}$ is cyclically reduced in
$G_{\alpha}$ and we come to the first alternative of Proposition 11.5.
Otherwise, according to Proposition 11.5α-1 we can assume that $Y$ is strongly
cyclically reduced in $G_{\alpha-1}$. Then we apply Lemma 11.7 to find a
strongly cyclically reduced in $G_{\alpha}$ word $Z$ conjugate to $Y$ in
$G_{\alpha}$ (note that $\rho+2\lambda+16\omega<1-4\lambda-8\omega$), coming
to the second alternative.

Let $\bar{\mathsf{Y}}=\prod_{i\in\mathbb{Z}}\mathsf{Y}_{i}$ and
$\prod_{i\in\mathbb{Z}}\mathsf{X}_{1}^{(i)}\mathsf{v}_{1}^{(i)}\mathsf{P}_{i}\mathsf{v}_{2}^{(i)}$
be lines in $\Gamma_{\alpha-1}$ representing the conjugacy relation. We
observe that

  1. (i)

The base axis of any fragment $\mathsf{N}$ of rank $\alpha$ in
$\mathsf{P}_{i}$ with $\mu_{\mathrm{f}}(\mathsf{N})\geq\lambda$ is the
infinite periodic extension of $\mathsf{P}_{i}$. In particular, If
$\mathsf{N}_{1}$ and $\mathsf{N}_{2}$ are fragments of rank $\alpha$ in
$\mathsf{P}_{i}$ with $\mu_{\mathrm{f}}(\mathsf{N}_{j})\geq\lambda$ then
$\mathsf{N}_{1}\sim\mathsf{N}_{2}$. (This follows from Corollary 8.2.)

Now formulate some consequences of the choice of $X_{1}$ of minimal possible
length:

  1. (ii)

There exist no fragments $\mathsf{N}_{1}$ and $\mathsf{N}_{2}$ of rank
$\alpha$ in $\mathsf{X}_{1}^{(i)}$ and in $\mathsf{X}_{1}^{(i+1)}$,
respectively, such that $\mathsf{N}_{1}\sim\mathsf{N}_{2}$ and
$\mu_{\mathrm{f}}(\mathsf{N}_{i})\geq 3.2\omega$.

Indeed, assume that such $\mathsf{N}_{1}$ and $\mathsf{N}_{2}$ do exist. Note
that both $\mathsf{N}_{1}$ and $\mathsf{N}_{2}$ are nonempty by Lemma 10.8. By
Lemma 10.13(i), any two of the endpoints of the images of $\mathsf{N}_{1}$ and
$\mathsf{N}_{2}$ in $\Gamma_{\alpha}$ are close. Then we can shorten $X_{1}$
to its subword $X_{2}$ so that $X_{2}u^{\prime}$ is conjugate to $X$ in
$G_{\alpha}$ for some $u^{\prime}\in\mathcal{H}_{\alpha}$ contrary to the
choice of $X_{1}$ (see Figure 35a; in the figure we have $\mathsf{N}_{2}\ll
s_{Y,\bar{\mathsf{Y}}}\mathsf{N}_{1}$ in $\mathsf{X}_{1}^{(i+1)}$ but in all
other cases we can easily find an appropriate path $\mathsf{X}_{2}$ with
$|\mathsf{X}_{2}|<\mathsf{X}_{1}$ and take
$X_{2}:=\operatorname{\mathit{label}}(\mathsf{X}_{2})$).

$\mathsf{X}_{1}^{(i+1)}$
---
$\mathsf{N}_{1}$
---
$\mathsf{N}_{2}$
---
$s_{Y,\bar{\mathsf{Y}}}\mathsf{N}_{1}$
---
$\mathsf{X}_{2}$
---
$\mathsf{X}_{1}^{(i)}$
---
$\mathsf{X}_{1}^{(i)}$
---
$\mathsf{X}_{1}^{(i+1)}$
---
$\mathsf{N}_{1}$
---
$\mathsf{N}_{2}$
---
$\mathsf{P}_{i}$
---
$\mathsf{P}_{i}$
---
$\mathsf{v}_{1}^{(i)}$
---
$\mathsf{v}_{2}^{(i)}$
---
$\mathsf{v}_{1}^{(i)}$
---
$\mathsf{v}_{2}^{(i)}$
---
a
---
b
---
Figure 35.

  1. (iii)

There exist no fragments $\mathsf{N}_{1}$ and $\mathsf{N}_{2}$ of rank
$\alpha$ in $\mathsf{X}_{1}^{(i)}$ and in $\mathsf{P}_{i}$ or
$\mathsf{P}_{i-1}$, respectively, such that
$\mathsf{N}_{1}\sim\mathsf{N}_{2}$, $\mu_{\mathrm{f}}(\mathsf{N}_{1})\geq
3.2\omega$ and $\mu_{\mathrm{f}}(\mathsf{N}_{2})\geq\lambda$. (Otherwise using
(i) we can shorten $X_{1}$ to
$X_{2}:=\operatorname{\mathit{label}}(\mathsf{X}_{2})$ as shown in Figure
35b.)

Let $Q$ be a word reduced in $G_{\alpha-1}$ which is equal to $X_{1}v_{1}P$ in
$G_{\alpha-1}$. We denote $\mathsf{Q}_{i}$ the corresponding path in
$\Gamma_{\alpha-1}$ joining $\iota(\mathsf{X}_{1}^{(i)})$ with
$\tau(\mathsf{P}_{i})$. Using (iii), Proposition 8.8 and Lemma 11.9 we
conclude that

  1. (iv)

There are no fragments $\mathsf{M}$ of rank $\alpha$ in $\mathsf{Q}_{i}$ with
$\mu_{\mathrm{f}}(\mathsf{M})\geq\rho+\lambda+6.2\omega$.

Assume that $\mathsf{K}$ is a fragment of rank $\alpha$ in $\bar{\mathsf{Y}}$
with $\mu_{\mathrm{f}}(\mathsf{K})\geq\rho+2\lambda+16\omega$ and
$|\mathsf{K}|\leq|Y|$. By (iv) and Proposition 8.9, for some $i$ there are
fragments $\mathsf{M}_{1}$ and $\mathsf{M}_{2}$ of rank $\alpha$ in
$\mathsf{Q}_{i}$ and $\mathsf{Q}_{i+1}$ respectively such that
$\mathsf{M}_{j}\sim\mathsf{K}$ $(i=1,2)$ and
$\mu_{\mathrm{f}}(\mathsf{M}_{j})>\lambda+6.8\omega$. By Proposition 8.8 there
is a fragment $\mathsf{N}_{1}$ of rank $\alpha$ such that
$\mathsf{M}_{1}\sim\mathsf{N}_{1}$ and either $\mathsf{N}_{1}$ occurs in
$\mathsf{X}_{1}^{(i)}$ and $\mu_{\mathrm{f}}(\mathsf{N}_{1})>3.2\omega$ or
$\mathsf{N}_{1}$ occurs in $\mathsf{P}_{i}$ and
$\mu_{\mathrm{f}}(\mathsf{N}_{1})>\lambda$. Similarly, there is a fragment
$\mathsf{N}_{2}$ of rank $\alpha$ such that $\mathsf{M}_{2}\sim\mathsf{N}_{2}$
and either $\mathsf{N}_{2}$ occurs in $\mathsf{X}_{1}^{(i+1)}$ and
$\mu_{\mathrm{f}}(\mathsf{N}_{2})>3.2\omega$ or $\mathsf{N}_{2}$ occurs in
$\mathsf{P}_{i+1}$ and $\mu_{\mathrm{f}}(\mathsf{N}_{2})>\lambda$. If
$\mathsf{N}_{1}$ occurs in $\mathsf{X}_{1}^{(i)}$ and $\mathsf{N}_{2}$ occurs
in $\mathsf{X}_{1}^{(i+1)}$ we get a contradiction with (ii). If
$\mathsf{N}_{1}$ occurs in $\mathsf{P}_{i}$ and $\mathsf{N}_{2}$ occurs in
$\mathsf{X}_{1}^{(i+1)}$ or $\mathsf{N}_{1}$ occurs in $\mathsf{X}_{1}^{(i)}$
and $\mathsf{N}_{2}$ occurs in $\mathsf{P}_{i+1}$ we get a contradiction with
(iii). Finally, if $\mathsf{N}_{1}$ occurs in $\mathsf{P}_{i}$ and
$\mathsf{N}_{2}$ occurs in $\mathsf{P}_{i+1}$ then by (i), we have
$s_{Y,\bar{\mathsf{Y}}}\mathsf{N}_{1}\sim\mathsf{N}_{2}$ and hence
$\mathsf{K}\sim s_{Y,\bar{\mathsf{Y}}}\mathsf{K}$. By Proposition 8.16(i)α-1
this implies that $Y$ is conjugate in $G_{\alpha-1}$ to a power of the root of
a relator of rank $\alpha$. This finishes the proof. ∎

######  11.11 Proposition.

Let $R$ be a relator of rank $\beta\leq\alpha$ and let $R\eqcirc R_{0}^{n}$
where $R_{0}$ is the root of $R$. Then $R_{0}$ has order $n$ in $G_{\alpha}$.

###### Proof.

Let $k$ be a proper divisor of $n$. By Lemma 10.8, $R_{0}^{k}$ contains no
fragments $K$ of rank $\gamma$ with $\mu_{\mathrm{f}}(K)\geq 3.2\omega$, for
all $\gamma=\beta+1,\dots,\alpha$. By Proposition 11.10β, $R_{0}^{k}$ is
cyclically reduced in $G_{\beta}$ and hence also in rank $\alpha$. Hence
$R_{0}^{k}\neq 1$ in $G_{\alpha}$. ∎

######  11.12 Proposition (conjugate powers of relator roots).

Let $R$ be a relator of rank $1\leq\beta\leq\alpha$ and let $R\eqcirc
R_{0}^{n}$ where $R_{0}$ is the root of $R$. If $R_{0}^{k}=g^{-1}R_{0}^{l}g$
in $G_{\alpha}$ for some $k,l\not\equiv 0\pmod{n}$ then $g\in\langle
R_{0}\rangle$ and $k\equiv l\pmod{n}$.

###### Proof.

By Proposition 11.11, if $R_{0}^{k}=g^{-1}R_{0}^{l}g$ in $G_{\alpha}$ and
$g\in\langle R_{0}\rangle$ then $k\equiv l\pmod{n}$. It remains to prove that
equality $R_{0}^{k}=g^{-1}R_{0}^{l}g$ for $k,l\not\equiv 0\pmod{n}$ implies
$g\in\langle R_{0}\rangle$.

By Proposition 11.10 we can assume that $R_{0}^{k}$ and $R_{0}^{l}$ are
cyclically reduced in $G_{\alpha}$. We represent $g$ by a word $Z$ and
consider an annular diagram $\Delta$ of rank $\alpha$ with two cyclic sides
$\mathsf{X}_{1}$ and $\mathsf{X}_{2}$ labeled $R_{0}^{-k}$ and $R_{0}^{l}$
which is obtained from a disk diagram with boundary label
$R_{0}^{-k}Z^{-1}R_{0}^{l}Z$ by gluing two boundary segments labeled $Z^{-1}$
and $Z$. Let $\mathsf{Z}$ be the path in $\Delta$ with
$\operatorname{\mathit{label}}(\mathsf{Z})\eqcirc Z$ that joins starting
vertices of $\mathsf{X}_{2}$ and $\mathsf{X}_{1}$.

We apply to $\Delta$ the reduction process 5.7. By Lemma 4.8, we can replace
$\mathsf{Z}$ by a new path $\mathsf{Z}_{1}$ with the same endpoints such that
$\operatorname{\mathit{label}}(Z_{1})=Z$ in $G_{\alpha}$ (so
$\operatorname{\mathit{label}}(Z_{1})$ represents $g$ in $G_{\alpha}$). We can
assume also that $\Delta$ has a tight set $\mathcal{T}$ of contiguity
subdiagrams.

Case 1: $\Delta$ has a cell $\mathsf{D}$ of rank $\alpha$. By Proposition
7.13(i), $\mathsf{D}$ has a contiguity subdiagram $\Pi_{i}\in\mathcal{T}$ to
each of the sides $\mathsf{X}_{i}$ of $\Delta$. Moreover, if
$\delta\Pi_{i}=\mathsf{S}_{i}\mathsf{u}_{i}\mathsf{Q}_{i}\mathsf{v}_{i}$ where
$\mathsf{S}_{i}^{-1}$ is a contiguity arc occurring in $\delta\mathsf{D}$ then
$\mu(\mathsf{S}_{i})>\lambda$. By Lemma 10.8 this implies $\beta=\alpha$. Let
$\operatorname{\mathit{label}}(\delta\Delta)\eqcirc R^{\prime}$ where
$R^{\prime}$ is a relator of rank $\alpha$. Consider lines
$\bar{\mathsf{X}}_{1}$, $\bar{\mathsf{X}}_{2}$ and $\bar{\mathsf{R}}$ in
$\Gamma_{\alpha-1}$ labeled $R^{\pm\infty}$, $R^{\pm\infty}$ and
$R^{\prime\infty}$ which are obtained by mapping the universal cover of the
subgraph of $\Delta$ shown in Figure 36.

$\mathsf{X}_{1}$
---
$\mathsf{X}_{2}$
---
$\mathsf{Z}_{1}$
---
$\mathsf{S}_{1}$
---
$\mathsf{Q}_{1}$
---
$\mathsf{u}_{1}$
---
$\mathsf{v}_{1}$
---
$\mathsf{S}_{2}$
---
$\mathsf{u}_{2}$
---
$\mathsf{Q}_{2}$
---
$\mathsf{v}_{2}$
---
$\mathsf{R}$
---
$\bar{\mathsf{X}}_{1}$
---
$\bar{\mathsf{X}}_{2}$
---
$\mathsf{D}$
---
Figure 36.

By Corollary 8.2 we get
$\bar{\mathsf{X}}_{1}=\bar{\mathsf{X}}_{2}=\bar{\mathsf{R}}$. This implies
that $\operatorname{\mathit{label}}(\mathsf{Z}_{1})$ is equal in
$G_{\alpha-1}$ to a power of $R_{0}$, as required.

Case 2: $\Delta$ has no cells of rank $\alpha$. Then we have equality
$R_{0}^{k}=Z_{1}^{-1}R_{0}^{l}Z_{1}$ in $G_{\alpha-1}$. If $\beta<\alpha$ then
the statement follows from Proposition 11.12α-1. Let $\beta=\alpha$. If $kl>0$
then the statement follows from Proposition 13.8α-1. If $kl<0$ then by
Corollary 13.10(i)α-1 we obtain $R_{0}=g^{-1}R_{0}^{-1}g$ which contradicts
our condition (S3) on the presentation of $G_{\alpha}$. ∎

######  11.13 Proposition.

Every element of $G_{\alpha}$ of infinite order has the form $h^{m}$ where $h$
is a non-power.

###### Proof.

We need to prove this only in the case $\alpha\geq 1$. Let $g\in G_{\alpha}$
be an element of infinite order. It is enough to find an upper bound on $|m|$
in the equality of the form $g=h^{m}$. Up to conjugation, we represent $g$ and
$h$ by a strongly cyclically reduced in $G_{\alpha}$ words $X$ and $Y$ by
Proposition 11.5. Let $\beta$ be the maximal rank with $1\leq\beta\leq\alpha$
such that a cyclic shift of $X$ contains a fragment $K$ of rank $\beta$ with
$\mu_{\mathrm{f}}(K)\geq\xi_{0}$. (It there is no such $K$ then by Proposition
9.16 $X$ in conjugate to $Y^{m}$ in the free group $G_{0}$ and then
$|m|\leq|X|$.) Using Propositions 10.24(i) and 8.16(ii) we find $m$ pairwise
non-compatible fragments $M$ of rank $\beta$ with
$\mu_{\mathrm{f}}(M)\geq\xi_{0}-2\lambda-3.4\omega$ in a cyclic shift of $X$.
This again implies $|m|\leq|X|$. ∎

##  12\. Coarsely periodic words and segments over $G_{\alpha}$

In this section we analyze words which are ‘‘geometrically close’’ in
$G_{\alpha}$ to periodic words. In Sections 12 and 13 we use the following
notation for numeric parameters:

$\xi_{1}=\xi_{0}-2.6\omega,\quad\xi_{2}=\xi_{1}-2\lambda-3.4\omega.$

######  12.1 Definition.

A simple period over $G_{\alpha}$ is a strongly cyclically reduced word
representing a non-power element of $G_{\alpha}$.

According to 2.5, if $A$ is a simple period over $G_{\alpha}$ then any word
$A^{n}$ is reduced over $G_{\alpha}$. Proposition 7.6 implies that $A$ has
infinite order in $G_{\alpha}$.

######  12.2 Definition.

Let $A$ be a simple period over $G_{\alpha}$. The activity rank of $A$ is the
maximal rank $\beta$ such that an $A$-periodic word contains a fragment $K$ of
rank $\beta\geq 1$ with $\mu_{\mathrm{f}}(K)\geq\xi_{1}$ or it is $0$ if no
such fragments exist.

###  12.3. Case of activity rank 0

The arguments below differ depending on whether the activity rank $\beta$ of a
simple period over $G_{\alpha}$ is positive or $0$. However, the difference is
only that in the case $\beta\geq 1$ we use various conditions on the size
$\mu_{\mathrm{f}}(\mathsf{F})$ of fragments $\mathsf{F}$ of rank $\beta$. All
definitions, statements and proofs in Sections 12 and 13 apply in cases when
the activity rank $\beta$ of a simple period over $G_{\alpha}$ is 0 simply
ignoring conditions of the form $\mu_{\mathrm{f}}(\cdot)\geq\dots$ for
fragments of rank $\beta$ (i.e. assuming that these conditions are all
formally true in case $\beta=0$). Below we do not distinguish this special
case $\beta=0$.

We will use the following notations. If $\mathsf{K}$ and $\mathsf{M}$ are
fragments of the same rank $0\leq\beta\leq\alpha$ occurring in a reduced path
$\mathsf{X}$ in $\Gamma_{\gamma}$ then $\mathsf{K}\lesssim\mathsf{M}$ means
$\mathsf{K}<\mathsf{M}$ or $\mathsf{K}\sim\mathsf{M}$; similarly,
$\mathsf{K}\lnsim\mathsf{M}$ means $\mathsf{K}<\mathsf{M}$ and
$\mathsf{K}\not\sim\mathsf{M}$ . Note that by Corollary 9.24(ii), for
fragments $\mathsf{K}$, $\mathsf{M}$ of rank $\beta\geq 1$ with
$\mu_{\mathrm{f}}(\mathsf{K}),\mu_{\mathrm{f}}(\mathsf{L})\geq\gamma+2.6\omega$
the relation ‘$\mathsf{K}\lesssim\mathsf{M}$’ depends only on their
equivalence classes with respect to compatibility. Thus, for fixed
$\mathsf{X}$ and $\beta$ it induces the linear order on the set of equivalence
classes of ‘$\sim$’ of fragments $\mathsf{N}$ of rank $\beta$ in $\mathsf{X}$
with $\mu_{\mathrm{f}}(\mathsf{N})\geq\gamma+2.6\omega$. (In case $\beta=0$
relation $\mathsf{K}\lesssim\mathsf{M}$ is defined on subpaths on length 1 and
means $\mathsf{K}\ll\mathsf{M}$ or $\mathsf{K}=\mathsf{M}$.)

######  12.4 Definition.

Let $A$ be a simple period over $G_{\alpha}$ and $\beta$ the activity rank of
$A$. A reduced path $\mathsf{S}$ in $\Gamma_{\alpha}$ is a coarsely periodic
segment with period $A$ (or a coarsely $A$-periodic segment for short) if
there exists a path $\mathsf{P}$ labeled by an $A$-periodic word, fragments
$\mathsf{K}_{0}$, $\mathsf{K}_{1}$ of rank $\beta$ in $\mathsf{P}$ and
fragments $\mathsf{M}_{0}$, $\mathsf{M}_{1}$ of rank $\beta$ in $\mathsf{S}$
such that:

  * •

$\mathsf{P}$ starts with $\mathsf{K}_{0}$ and ends with $\mathsf{K}_{1}$;
$\mathsf{S}$ starts with $\mathsf{M}_{0}$ and ends with $\mathsf{M}_{1}$;

  * •

$\mathsf{K}_{0}\sim\mathsf{M}_{0}^{\pm 1}$,
$\mathsf{K}_{1}\sim\mathsf{M}_{1}^{\pm 1}$ and
$\mathsf{K}_{0}\not\sim\mathsf{K}_{1}$;

  * •

$\mu_{\mathrm{f}}(\mathsf{K}_{i})\geq\xi_{1}$,
$\mu_{\mathrm{f}}(\mathsf{M}_{i})\geq\xi_{2}$ ($i=0,1$);

  * •

$s_{A,\mathsf{P}}\mathsf{K}_{0}\lesssim\mathsf{K}_{1}$ (informally,
$\mathsf{P}$ ‘‘contains at least one period $A$’’).

The path $\mathsf{P}$ is a periodic base for $\mathsf{S}$. The infinite
$A$-periodic extension of $\mathsf{P}$ is an axis for $\mathsf{S}$.

Note that the starting fragment $\mathsf{M}_{0}$ and the ending fragment
$\mathsf{M}_{1}$ of $\mathsf{S}$ are defined up to compatibility.

Note also that by Lemma 10.13(i) and Proposition 9.10, $\mathsf{P}$ and
$\mathsf{S}$ are close in rank $\beta$. In particular, if $\beta=0$ then
$\mathsf{P}=\mathsf{Q}$ and thus $\mathsf{P}$ is an $A$-periodic segment.

We will be assuming that a coarsely $A$-periodic segment is always considered
with a fixed associated axis. (In fact, we prove later that the axis of a
coarsely $A$-periodic segment is defined in a unique way, see Corollary 13.9).
Note that under this assumption, the periodic base $\mathsf{P}$ for
$\mathsf{S}$ is defined up to changing the starting and the ending fragments
$\mathsf{K}_{0}$ and $\mathsf{K}_{1}$ of rank $\beta$ with compatible ones.

The label of a coarsely $A$-periodic segment in $\Gamma_{\alpha}$ is a
coarsely $A$-periodic word over $G_{\alpha}$.

Note that a simple period $A$ over $G_{0}$ is any cyclically freely reduced
word that is not a proper power. A coarsely $A$-periodic word over $G_{0}$ is
simply any $A$-periodic word $P$ with $|P|>|A|$.

######  12.5 Definition.

We measure the size of a coarsely $A$-periodic segment $\mathsf{S}$, which
roughly corresponds to the number of periods $A$, in the following way. Let
$\mathsf{P}$ be the periodic base for $\mathsf{S}$ and $\mathsf{K}_{0}$,
$\mathsf{K}_{1}$ as in Definition 12.4. Then we write $\ell_{A}(\mathsf{S})=t$
where $t$ is the maximal integer such that
$s_{A,\mathsf{P}}^{t}\mathsf{K}_{0}\lesssim\mathsf{K}_{1}$. Thus, we always
have $\ell_{A}(\mathsf{S})\geq 1$.

Since we consider a fixed associated axis for $\mathsf{S}$, the number
$\ell_{A}(\mathsf{S})$ does not depend on the choice of a periodic base
$\mathsf{P}$.

If $S$ is a coarsely $A$-periodic word over $G_{\alpha}$ then we formally
define $\ell_{A}(S)$ to be the maximal possible value of
$\ell_{A}(\mathsf{S})$ where $\mathsf{S}$ is a coarsely $A$-periodic segment
labeled $S$.

######  12.6 Remark.

(i) It immediately follows from the definition that $t$ is also the maximal
integer such that $\mathsf{K}_{0}\lesssim
s_{A,\mathsf{P}}^{-t}\mathsf{K}_{1}$. Thus,
$\ell_{A}(\mathsf{S})=\ell_{A^{-1}}(\mathsf{S}^{-1})$.

(ii) To compute $\ell_{A}(S)$ we have to take a path $\mathsf{S}$ in
$\Gamma_{\alpha}$ with $\operatorname{\mathit{label}}(\mathsf{S})\eqcirc S$
and then choose a periodic base $\mathsf{P}$ for $\mathsf{S}$ so that
$\ell_{A}(\mathsf{S})$ is maximal possible; it will follow from Proposition
13.7 that any choice of $\mathsf{P}$ gives in fact the same value
$\ell_{A}(\mathsf{S})$.

######  12.7 Remark.

Up to changing the periodic base $\mathsf{P}$, we can always assume in
Definition 12.5 that both $\mathsf{K}_{0}$ and its translation
$s_{A,\mathsf{P}}^{t}\mathsf{K}_{0}$ occur in $\mathsf{P}$. In this case we
have $|\mathsf{P}|\geq\ell_{A}(\mathsf{S})|A|$.

######  12.8 Definition.

Let $\mathsf{S}_{1}$ and $\mathsf{S}_{2}$ be coarsely $A$-periodic segments in
$\Gamma_{\alpha}$.

We say that $\mathsf{S}_{1}$ and $\mathsf{S}_{2}$ are compatible if they have
the same axis and strongly compatible if they share a common periodic base.

We use notations $\mathsf{S}_{1}\sim\mathsf{S}_{2}$ and
$\mathsf{S}_{1}\approx\mathsf{S}_{2}$ for compatibility and strong
compatibility respectively.

Note that in the case $\mathsf{S}_{1}\approx\mathsf{S}_{2}$ any periodic base
for $\mathsf{S}_{1}$ is a periodic base for $\mathsf{S}_{2}$ and vice versa.
This easily follows from Definition 12.4.

If $\mathsf{S}_{1}$ and $\mathsf{S}_{2}$ are coarsely $A$-periodic segments in
$\Gamma_{0}$ then $\mathsf{S}_{1}\sim\mathsf{S}_{2}$ if and only if they have
a common periodic extension and $\mathsf{S}_{1}\approx\mathsf{S}_{2}$ if and
only if $\mathsf{S}_{1}=\mathsf{S}_{2}$.

######  12.9 Proposition.

Let $\mathsf{S}_{1}$ and $\mathsf{S}_{2}$ be coarsely $A$-periodic segments in
$\Gamma_{\alpha}$.

  1. (i)

If $\mathsf{S}_{1}\approx\mathsf{S}_{2}$ then
$\ell_{A}(\mathsf{S}_{1})=\ell_{A}(\mathsf{S}_{2})$.

  2. (ii)

Assume that $\mathsf{S}_{1}$ and $\mathsf{S}_{2}$ occur in a reduced path
$\mathsf{X}$ in $\Gamma_{\alpha}$ and $\mathsf{S}_{1}\sim\mathsf{S}_{2}$. Then
the union of $\mathsf{S}_{1}$ and $\mathsf{S}_{2}$ in $\mathsf{X}$ is an
$A$-coarsely periodic segment where a periodic base for
$\mathsf{S}_{1}\cup\mathsf{S}_{2}$ is the union of periodic bases f or
$\mathsf{S}_{1}$ and $\mathsf{S}_{2}$ in their common infinite $A$-periodic
extension.

###### Proof.

(i) is immediate consequence of Definition 12.8.

(ii) follows from Proposition 10.23(ii). ∎

### 12.10.

We describe a procedure of shortening a coarsely $A$-periodic segment
$\mathsf{S}$ by a ‘‘given number $k$ of periods’’. Let $k\geq 1$ and
$\ell_{A}(\mathsf{S})\geq k+1$. Let $\beta$ be the activity rank of
$\mathsf{S}$, let $\mathsf{P}$ a periodic base for $\mathsf{S}$ and let
$\mathsf{K}_{i}$ and $\mathsf{M}_{i}$ $(i=0,1)$ be starting and ending
fragments of rank $\beta$ of $\mathsf{P}$ and $\mathsf{S}$ respectively as in
Definition 13.3. We have
$\mathsf{K}_{0}<s_{A,\mathsf{P}}^{k}\mathsf{K}_{0}\lesssim
s_{A,\mathsf{P}}^{-1}\mathsf{K}_{1}<\mathsf{K}_{1}$ and it follows from
Proposition 8.16(ii) that
$s_{A,\mathsf{P}}^{k}\mathsf{K}_{0}\not\sim\mathsf{K}_{0}$ and
$s_{A,\mathsf{P}}^{k}\mathsf{K}_{0}\not\sim\mathsf{K}_{1}$. By Proposition
10.23(i) there exists a fragment $\mathsf{N}$ of rank $\beta$ in $\mathsf{S}$
with $\mu_{\mathrm{f}}(\mathsf{N}^{\prime})\geq\xi_{2}$ such that
$s_{A,\mathsf{P}}^{k}\mathsf{K}_{0}\sim\mathsf{N}^{\pm 1}$. Then
$\mathsf{S}_{1}=\mathsf{N}\cup\mathsf{M}_{1}$ is an end of $\mathsf{S}$ which
is a coarsely $A$-periodic segment with periodic base
$\mathsf{P}_{1}=s_{A,\mathsf{P}}^{k}\mathsf{K}_{0}\cup\mathsf{K}_{1}$ and
$\ell_{A}(\mathsf{S}_{1})=\ell_{A}(\mathsf{S})-k$. We note that:

  1. (i)

The result of the operation is defined up to the strict compatibility.

  2. (ii)

We have $\mathsf{P}=\mathsf{X}\mathsf{P}_{1}$ where $|\mathsf{X}|=k|A|$.

  3. (iii)

If $k\geq 2$ then by Proposition 10.23(i) we can find also a fragment
$\mathsf{N}^{\prime}$ of rank $\beta$ in $\mathsf{S}$ with
$\mu_{\mathrm{f}}(\mathsf{N}^{\prime})\geq\xi_{2}$ such that
$s_{A,\mathsf{P}}^{k-1}\mathsf{K}_{0}\sim\mathsf{N}^{\prime\pm 1}$ and
$\mathsf{N}^{\prime}$ and $\mathsf{N}$ are disjoint. Then
$\mathsf{S}=\mathsf{S}_{0}\mathsf{u}\mathsf{S}_{1}$ where
$\mathsf{S}_{0}=\mathsf{M}_{0}\cup\mathsf{N}^{\prime}$ is a coarsely
$A$-periodic segment with periodic base $\mathsf{K}_{0}\cup
s_{A,\mathsf{P}}^{k-1}\mathsf{K}_{0}$ and $\ell_{A}(\mathsf{S}_{0})=k-1$.

  4. (iv)

The starting position of $\mathsf{S}_{1}$ depends only on the starting
position of $\mathsf{S}$; more precisely, if $\mathsf{S}^{\prime}$ is a start
of $\mathsf{S}$ and $\mathsf{S}_{1}$ and $\mathsf{S}_{1}^{\prime}$ are
obtained from $\mathsf{S}$ and $\mathsf{S}^{\prime}$ as above then
$\mathsf{S}_{1}^{\prime}$ is a start of $\mathsf{S}_{1}$ up to strict
compatibility of $\mathsf{S}_{1}^{\prime}$; if
$\mathsf{S}\approx\mathsf{S}^{\prime}$ then
$\mathsf{S}_{1}\approx\mathsf{S}_{1}^{\prime}$.

######  12.11 Definition.

If $\mathsf{S}_{1}$ is obtained from $\mathsf{S}$ by the procedure in 12.10
then we say that $\mathsf{S}_{1}$ is obtained by shortening of $\mathsf{S}$ by
$t$ periods from the start. In the symmetric way, we define shortening of
$\mathsf{S}$ by $t$ periods from the end.

If $\ell_{A}(\mathsf{S})\geq 2t+1$ and $\mathsf{S}^{\prime}$ is obtained from
$\mathsf{S}$ by applying the operation from both sides then
$\mathsf{S}^{\prime}$ is the result of truncation of $\mathsf{S}$ by $t$
periods.

######  12.12 Definition.

We define two numeric parameters associated with a simple period $A$ over
$G_{\alpha}$: the stable size $[A]_{\alpha}$ of $A$ in rank $\alpha$,

$[A]_{\alpha}=\inf_{m\geq 1}\frac{|(A^{m})^{\circ}|_{\alpha}}{m}$

and the stability decrement $h_{\alpha}(A)$:

$h_{\alpha}(A)=\left\lceil\frac{1.2}{[A]_{\alpha}}\right\rceil+1.$

If $\ell_{A}(\mathsf{S})\geq 2h_{\alpha}(A)+1$ then the result of truncation
of $\mathsf{S}$ by $h_{\alpha}(A)$ periods is the stable part of $\mathsf{S}$.
By claim 12.10(iv) and its symmetric version, the function
‘$\mathsf{S}\to\text{stable part of }\mathsf{S}$’ respects strict
compatibility: if $\mathsf{S}_{1}\approx\mathsf{S}_{2}$ and
$\mathsf{S}_{i}^{*}$ is the stable part of $\mathsf{S}_{i}$ then
$\mathsf{S}_{1}^{*}\approx\mathsf{S}_{2}^{*}$.

The basic fact about $[A]_{\alpha}$ and $h_{\alpha}(A)$ is the following
observation.

######  12.13 Lemma.

If $X$ is an $A$-periodic word and $|X|\geq m|A|$ then $|X|_{\alpha}\geq
m[A]_{\alpha}$. In particular, if $|X|\geq(h_{\alpha}(A)-1)|A|$ then
$|X|_{\alpha}\geq 1.2$.

###### Proof.

We have

$|X|_{\alpha}\geq|A_{1}^{m}|_{\alpha}\geq|(A^{m})^{\circ}|_{\alpha}\geq
m[A]_{\alpha}$

where $A_{1}$ is the cyclic shift of $A$ at which $X$ starts. The second
statement follows from the first. ∎

The principal role of the stable part is described by the following
proposition.

######  12.14 Proposition (stability of coarsely periodic words).

Let $\mathsf{S}$ be a coarsely $A$-periodic segment in $\Gamma_{\alpha}$ with
$\ell_{A}(\mathsf{S})\geq 2h_{\alpha}(A)+1$ and let $\mathsf{S}^{*}$ be the
stable part of $\mathsf{S}$. If $\mathsf{X}$ and $\mathsf{Y}$ are close
reduced paths in $\Gamma_{\alpha}$ and $\mathsf{S}$ is a subpath of
$\mathsf{X}$ then $\mathsf{Y}$ contains a coarsely $A$-periodic segment
$\mathsf{T}$ such that $\mathsf{T}\approx\mathsf{S}^{*}$.

###### Proof.

Let $\mathsf{P}$ and $\mathsf{P}^{*}$ be periodic bases for $\mathsf{S}$ and
$\mathsf{S}^{*}$ respectively. Let $\beta$ be the activity rank of $A$ and let
$\mathsf{K}_{i}$ and $\mathsf{M}_{i}$ $(i=0,1)$ be fragments of rank $\beta$
in $\mathsf{P}$ and in $\mathsf{S}$, respectively, from Definition 13.3
applied to $\mathsf{P}$ and $\mathsf{S}$. Denote $t=h_{\alpha}(A)$.

Let $\mathsf{X}$ and $\mathsf{Y}$ be as in the proposition. If $\alpha=0$ then
$\mathsf{X}=\mathsf{Y}$ and there is nothing to prove. Let $\alpha>0$. We
claim that $\mathsf{P}=\mathsf{z}_{1}\mathsf{P}^{\prime}\mathsf{z}_{2}$ where
$\mathsf{P}^{\prime}$ is close in rank $\beta$ to a subpath of $\mathsf{Y}$
and $|\mathsf{z}_{i}|_{\alpha}<1.2$. Indeed, if $\beta=\alpha$ then it easily
follows from Proposition 10.6 and Lemma 10.13(i) that $\mathsf{P}$ is already
close to a subpath of $\mathsf{Y}$. If $\beta<\alpha$ then we observe that
$\mathsf{S}$ contains no fragments $\mathsf{K}$ of rank $\gamma$ with
$\beta<\gamma\leq\alpha$ and $\mu_{\mathrm{f}}(\mathsf{K})\geq\xi_{0}$ due to
the definition of the activity rank and Proposition 8.7≤α. Then the claim
follows by Proposition 10.22.

By Lemma 12.13 we have $|\mathsf{z}_{i}|<(t-1)|A|$. This implies that
$s_{A,\mathsf{P}}^{t-1}\mathsf{K}_{0}\cup
s_{A,\mathsf{P}}^{-t+1}\mathsf{K}_{1}$ is contained in $\mathsf{P}^{\prime}$.
Note that $\mathsf{P}^{*}=s_{A,\mathsf{P}}^{t}\mathsf{K}_{0}\cup
s_{A,\mathsf{P}}^{-t}\mathsf{K}_{1}$ where
$\mu_{\mathrm{f}}(\mathsf{K}_{0}),\mu_{\mathrm{f}}(\mathsf{K}_{1})\geq\xi_{1}$.
Then by Proposition 10.23(i) we find a subpath $\mathsf{T}$ which is a
coarsely $A$-periodic segment with periodic base $\mathsf{P}^{*}$ and,
consequently, we have $\mathsf{T}\approx\mathsf{S}^{*}$. ∎

We use parameter $h_{\alpha}(A)$ also in several other situations.

######  12.15 Proposition.

Let $\mathsf{P}$ be a periodic segment in $\Gamma_{\alpha}$ with a simple
period $A$ over $G_{\alpha}$. Assume that $|\mathsf{P}|\geq m|A|$ where $m\geq
2h_{\alpha}(A)+3$. Let $\mathsf{X}$ be a reduced path in $\Gamma_{\alpha}$
such that $\mathsf{P}$ and $\mathsf{X}$ are close. Then there exist a subpath
$\mathsf{P}_{1}$ of $\mathsf{P}$ and a subpath $\mathsf{X}_{1}$ of
$\mathsf{X}$ such that $\mathsf{X}_{1}$ is a coarsely $A$-periodic segment
with periodic base $\mathsf{P}_{1}$ and
$\ell_{A}(\mathsf{X}_{1})=m-2h_{\alpha}(A)-2$.

###### Proof.

Let $\beta$ be the activity rank of $A$. Using Corollary 9.13 and Lemma 12.13
we find close in rank $\beta$ subpaths $\mathsf{P}_{2}$ of $\mathsf{P}$ and
$\mathsf{X}_{2}$ of $\mathsf{X}$ with $|\mathsf{P}_{2}|\geq
m-2h_{\alpha}(A)+2$. By Proposition 8.16(iii) any fragment $\mathsf{K}$ of
rank $\beta$ in $\mathsf{P}$ with $\mu_{\mathrm{f}}(\mathsf{K})\geq
2\lambda+5.3\omega$ satisfies $|\mathsf{K}|<2|A|$, so according to Definition
12.4 there exists a fragment $\mathsf{K}$ of rank $\beta$ in $\mathsf{P}$ with
$\mu_{\mathrm{f}}(\mathsf{K})\geq\xi_{1}$. Shortening $\mathsf{K}$ from the
end by Proposition 8.12 if $\beta\geq 1$ and using again Proposition 8.16(ii)
we find a fragment $\mathsf{K}_{1}$ of rank $\beta$ with
$\mu_{\mathrm{f}}(\mathsf{K}_{1})>\xi_{1}-\lambda-2.7\omega$ that is a start
of $\mathsf{K}$ disjoint from $s_{A,\mathsf{P}}\mathsf{K}$; hence
$|\mathsf{K}_{1}|\leq|A|$. We can assume that $\mathsf{K}$ occurs in
$\mathsf{P}_{2}$ and is closest to the start of $\mathsf{P}_{2}$. Then
$\mathsf{P}_{2}$ contains $m-2h_{\alpha}(A)$ translates
$s_{A,\mathsf{P}}^{i}\mathsf{K}$ of $\mathsf{K}$ for
$i=0,\dots,m-2h_{\alpha}(A)-1$ and contains also
$s_{A,\mathsf{P}}^{m-2h_{\alpha}(A)}\mathsf{K}_{1}$. Applying Proposition
10.23(i) we find fragments $\mathsf{M}_{i}$ $(i=1,\dots,m-2h_{\alpha}(A)-1)$
of rank $\beta$ in $\mathsf{X}_{2}$ with
$\mu_{\mathrm{f}}(\mathsf{M}_{i})\geq\xi_{2}$ such that
$s_{A,\mathsf{P}}^{i}\mathsf{K}\sim\mathsf{M}_{i}^{\pm 1}$. Then
$\mathsf{X}_{1}=\mathsf{M}_{1}\cup\mathsf{M}_{m-2h_{\alpha}(A)-1}$ is a
coarsely $A$-periodic segment with periodic base
$s_{A,\mathsf{P}}\mathsf{K}\cup
s_{A,\mathsf{P}}^{m-2h_{\alpha}(A)-1}\mathsf{K}$ and we have
$\ell_{A}(\mathsf{X}_{1})=m-2h_{\alpha}(A)-2$. ∎

######  12.16 Proposition.

Let $S$ be a coarsely $A$-periodic word over $G_{\alpha}$ and $B$ a simple
period over $G_{\alpha}$ conjugate to $A$. Let $\ell_{A}(S)\geq
2h_{\alpha}(A)+3$. Then a subword $T$ of $S$ is a coarsely $B$-periodic word
over $G_{\alpha}$ with $\ell_{B}(T)\geq\ell_{A}(S)-2h_{\alpha}(A)-2$.

###### Proof.

We represent $S$ by a coarsely $A$-periodic segment $\mathsf{S}$ in
$\Gamma_{\alpha}$. Let $\mathsf{P}$ a periodic base for $\mathsf{S}$, let
$\mathsf{L}_{1}$ be the axis of $\mathsf{S}$ and let $\mathsf{L}_{2}$ be the
$B$-periodic line parallel to $\mathsf{L}_{1}$. Denote $\beta_{1}$ and
$\beta_{2}$ activity ranks of $A$ and $B$ respectively.

According to Definition 12.2, either $\mathsf{L}_{1}$ or $\mathsf{L}_{2}$
contains no fragments $\mathsf{K}$ of rank $\gamma$ with
$\beta_{1}<\gamma\leq\alpha$ and $\mu_{\mathrm{f}}(\mathsf{K})\geq\xi_{1}$.
Let $\mathsf{K}_{0}$ and $\mathsf{K}_{1}$ be fragments of rank $\beta_{1}$
with $\mu_{\mathrm{f}}(\mathsf{K}_{i})\geq\xi_{1}$ that are a start and an end
of $\mathsf{P}$ respectively. We have
$s_{A,\mathsf{L}_{1}}^{\ell_{A}(\mathsf{S})}\mathsf{K}_{0}\lesssim\mathsf{K}_{1}$.
By Proposition 10.24(i), there exist fragments $\mathsf{M}_{0}$ and
$\mathsf{M}_{1}$ of rank $\beta_{1}$ in $\mathsf{L}_{2}$ with
$\mu_{\mathrm{f}}(\mathsf{M}_{i})\geq\xi_{2}$ such that
$\mathsf{K}_{i}\sim\mathsf{M}_{i}^{\pm 1}$. Since $\mathsf{L}_{1}$ and
$\mathsf{L}_{2}$ are parallel, we have
$s_{A,\mathsf{L}_{1}}=s_{B,\mathsf{L}_{2}}$ and hence
$s_{B,\mathsf{L}_{2}}^{\ell_{A}(\mathsf{S})}\mathsf{M}_{0}\lesssim\mathsf{M}_{1}$
by Proposition 10.24(ii). Then $\mathsf{Q}=\mathsf{M}_{0}\cup
s_{B,\mathsf{L}_{2}}^{\ell_{A}(\mathsf{S})}\mathsf{M}_{0}\cup\mathsf{M}_{1}$
is close in rank $\beta_{1}$ to $\mathsf{P}$,
$|\mathsf{Q}|\geq\ell_{A}(\mathsf{S})$ and the statement follows by
Proposition 12.15. ∎

##  13\. Overlapped coarse periodicity

The main result of this section is Proposition 13.4 which can be thought as an
analog of a well known property of periodic words: if two periodic words have
a sufficiently large overlapping then they have a common period. We need such
an analog in a more general context where closeness plays the role of
overlapping. As a main technical tool, instead of coincidence of letters in
the overlapping case we use correspondence of fragments of rank
$\beta\leq\alpha$ in strictly close in rank $\beta$ segments in
$\Gamma_{\alpha}$ given by Proposition 10.23. A difficulty is caused by the
‘‘fading effect’’ of this correspondence: a fragment size can decrease when
passing from one segment to the other. To overcome this difficulty, we use a
special combinatorial argument [9, Lemma 6.4].

######  13.1 Lemma (penetration lemma, [9, Lemma 6.4]).

Let $S_{0}$, $S_{1}$, $\dots$, $S_{k}$ be a finite collection of disjoint
sets. Assume that the following assertions hold:

  1. (i)

Each $S_{i}$ is pre-ordered, i.e. endowed with a transitive relation
‘$<_{i}$’.

  2. (ii)

There is an equivalence relation $a\sim b$ on the union $\bigcup_{i}S_{i}$
such that for any $a,b$ in the same set $S_{i}$ we have either $a<_{i}b$,
$b<_{i}a$ or $a\sim b$; in other words, we have an induced linear ordering on
the set of equivalence classes on each $S_{i}$.

  3. (iii)

We assume that the equivalence preserves the pre-ordering in neighboring sets:
if $a,b\in S_{i}$, $a^{\prime},b^{\prime}\in S_{i+1}$, $a\sim a^{\prime}$ and
$b\sim b^{\prime}$ then $a<_{i}b\Leftrightarrow a^{\prime}<_{i+1}b^{\prime}$.

If $c\in S_{i}$, $a,b\in S_{j}$ and $a\lesssim_{j}b$ (where $a\lesssim_{j}b$
denotes ‘$a<_{j}b$ or $a\sim b$’) then we say that $c$ penetrates between $a$
and $b$ if there exists $c^{\prime}\sim c$ such that
$a\lesssim_{j}c^{\prime}\lesssim_{j}b$.

  4. (iv)

There is a subset of $\bigcup_{i}S_{i}$ of stable elements that have the
following property: if $c\in S_{i}$ is stable, $a\lesssim_{i}c\lesssim_{i}b$,
$a^{\prime},b^{\prime}\in S_{j}$, $a^{\prime}\lesssim_{j}b^{\prime}$, $a\sim
a^{\prime}$ and $b\sim b^{\prime}$ then $c$ penetrates between $a^{\prime}$
and $b^{\prime}$.

  5. (v)

For each $i\leq k-1$, there are stable elements $a_{i},b_{i}\in S_{i}$ and
$a_{i}^{\prime},b_{i}^{\prime}\in S_{i+1}$ such that $a_{i}\sim
a_{i}^{\prime}$, $b_{i}\sim b_{i}^{\prime}$ and $a_{i}<_{i}b_{i}$.

Finally, let $c_{0}\in S_{0}$ be stable and
$a_{0}\lesssim_{0}c_{0}\lesssim_{0}b_{0}$. Assume that $c_{0}$ penetrates
between $a_{i}$ and $b_{i}$ for each $i=1,2,\dots,k-1$. Then $c_{0}$
penetrates between $a_{k}$ and $b_{k}$.

The following observation is a special case of [9, Lemma 6.2].

######  13.2 Lemma.

Suppose a group $G$ acts on set $X$. Let $g,h\in G$,
$x_{0},x_{1},\dots,x_{t}\in X$ and for some $r,s\geq 0$ with $\gcd(r,s)=1$ and
$r+s\leq t$,

$gx_{i}=x_{i+r}\ (i=0,1,\dots,t-r),\quad hx_{i}=x_{i+s}\ (i=0,1,\dots,t-s).$

Assume that the stabilizer $H$ of $x_{0}$ is malnormal in $G$. Then either
$g,h\in H$ (and hence $x_{0}=x_{1}=\dots=x_{t}$) or there exists $d\in G$ such
that $g=d^{r}$ and $h=d^{s}$.

###### Proof.

Induction on $r+s$. We can assume that $r\leq s$. If $r>0$ then we have
$g^{-1}hx_{i}=x_{i+s-r}$ for $0\leq i\leq t-s$ and the statement follows from
the inductive hypothesis with $h:=g^{-1}h$, $s:=s-r$ and $t:=t-r$. Otherwise
we have $r=0$ and $s=1$. Then $h^{-1}ghx_{0}=gx_{0}=x_{0}$ and by malnormality
of $H$, we have either $g,h\in H$ or $g=1$ (and then $g=h^{0}$ and $h=h^{1}$).
∎

######  13.3 Definition.

Let $\mathsf{X}$ and $\mathsf{Y}$ be reduced paths in $\Gamma_{\alpha}$. We
say that $\mathsf{X}$ and $\mathsf{Y}$ are strictly close in rank
$\beta\leq\alpha$ if there are fragments $\mathsf{K}_{0}$, $\mathsf{K}_{1}$ of
rank $\beta$ in $\mathsf{X}$ and fragments $\mathsf{M}_{0}$, $\mathsf{M}_{1}$
of rank $\beta$ in $\mathsf{Y}$ such that:

  * •

$\mu_{\mathrm{f}}(\mathsf{K}_{i}),\mu_{\mathrm{f}}(\mathsf{M}_{i})\geq\xi_{2}$
($i=0,1$).

  * •

$\mathsf{X}$ starts with $\mathsf{K}_{0}$ and ends with $\mathsf{K}_{1}$;
$\mathsf{Y}$ starts with $\mathsf{M}_{0}$ and ends with $\mathsf{M}_{1}$;

  * •

$\mathsf{K}_{0}\sim\mathsf{M}_{0}^{\pm 1}$,
$\mathsf{K}_{1}\sim\mathsf{M}_{1}^{\pm 1}$ and
$\mathsf{K}_{0}\not\sim\mathsf{K}_{1}$.

By Lemma 10.13(i), paths which are strictly close in rank $\beta$ are also
close in rank $\beta$. One of the advantages of strict closeness is that this
relation is transitive (this follows immediately from Definition 13.3). Note
that a coarsely periodic segment $\mathsf{P}$ in $\Gamma_{\alpha}$ and its
periodic base $\mathsf{S}$ are strictly close according to Definition 12.4
(and the condition in Definition 12.4 is slightly stronger because of the
lower bound on the size of the starting and the ending fragments of
$\mathsf{S}$).

######  13.4 Proposition.

Let $A$ be a simple period over $G_{\alpha}$, $\beta$ the activity rank of $A$
and $\mathsf{P}_{i}$ $(i=0,1)$ be two $A$-periodic segments in
$\Gamma_{\alpha}$. Let $\mathsf{S}_{i}$ $(i=0,1)$ be a reduced path in
$\Gamma_{\alpha}$ which is strictly close to $\mathsf{P}_{i}$. Assume that
$\mathsf{S}_{0}$ is contained in $\mathsf{S}_{1}$. Assume also that
$\mathsf{P}_{0}$ contains at least one period $A$ in the sense that there
exist fragments $\mathsf{K}$ and $\mathsf{K}^{\prime}$ of rank $\beta$ in
$\mathsf{P}_{0}$ such that
$\mu_{\mathrm{f}}(\mathsf{K}),\mu_{\mathrm{f}}(\mathsf{K}^{\prime})\geq\xi_{2}$
and $\mathsf{K}^{\prime}\sim s_{A,\mathsf{P}_{0}}\mathsf{K}$. Then
$\mathsf{P}_{0}$ and $\mathsf{P}_{1}$ have a common periodic extension.

###### Proof.

Denote

$\xi_{3}=\xi_{2}-2\lambda-3.4\omega=3\lambda-10.9\omega.$

Throughout the proof, ‘‘fragment $\mathsf{M}$’’ means ‘‘fragment $\mathsf{M}$
of rank $\beta$ with $\mu_{\mathrm{f}}(\mathsf{M})\geq\zeta_{3}$’’ (or simply
‘‘fragment $\mathsf{M}$ of rank 0’’ if $\beta=0$, see 12.3).

Let a line $\mathsf{L}_{i}$ be the infinite periodic extension of
$\mathsf{P}_{i}$ and let $g$ be an element of $G_{\alpha}$ such that
$\mathsf{L}_{1}=g\mathsf{L}_{0}$, so
$s_{A,\mathsf{P}_{1}}=gs_{A,\mathsf{P}_{0}}g^{-1}$. Our argument relies on
establishing a correspondence between fragments of rank $\beta$ in
$\mathsf{P}_{i}$ and $\mathsf{S}_{i}$. It will be convenient to consider
fragments of rank $\beta$ in four paths $\mathsf{P}_{i}$ and $\mathsf{S}_{i}$
as four disjoint sets, i.e. we will formally consider pairs
$(\mathsf{M},\mathsf{X})$ where
$\mathsf{X}\in\\{\mathsf{P}_{0},\mathsf{P}_{1},\mathsf{S}_{0},\mathsf{S}_{1}\\}$
and $\mathsf{M}$ is a fragment occurring in $\mathsf{X}$. We will refer to
$\mathsf{M}$ as a ‘‘fragment belonging to $\mathsf{X}$’’ or simply as a
‘‘fragment in $\mathsf{X}$’’.

We introduce two operations on fragments in $\mathsf{P}_{i}$ and
$\mathsf{S}_{i}$. Let $\mathsf{M}$ and $\mathsf{N}$ be fragments each
belonging to some $\mathsf{P}_{i}$ or $\mathsf{S}_{i}$.

  1. (i)

If $\mathsf{M}$ belongs to $\mathsf{P}_{i}$, $\mathsf{N}$ belongs to
$\mathsf{S}_{i}$ and $\mathsf{M}\sim\mathsf{N}^{\pm 1}$ then either of
$\mathsf{M}$ and $\mathsf{N}$ jumps to the other.

  2. (ii)

$\mathsf{M}$ translates to $\mathsf{N}$ in the following cases (a)–(d):

    1. (a)

$\mathsf{M}$ and $\mathsf{N}$ belong to the same $\mathsf{P}_{i}$ and
$\mathsf{N}\sim s_{A,\mathsf{P}_{i}}^{k}\mathsf{M}$ for some $k\in\mathbb{Z}$;
or

    2. (b)

$\mathsf{M}$ belongs to $\mathsf{P}_{0}$, $\mathsf{N}$ belongs to
$\mathsf{P}_{1}$ and $\mathsf{N}\sim gs_{A,\mathsf{P}_{0}}^{k}\mathsf{M}$ for
some $k\in\mathbb{Z}$; or

    3. (c)

$\mathsf{M}$ belongs to $\mathsf{P}_{1}$, $\mathsf{N}$ belongs to
$\mathsf{P}_{0}$ and $\mathsf{N}\sim g^{-1}s_{A,\mathsf{P}_{1}}^{k}\mathsf{M}$
for some $k\in\mathbb{Z}$.

(In other words, $\mathsf{M}$ translates to $\mathsf{N}$ in cases (a)–(c) if
they have the same position in their corresponding periodic lines
$\mathsf{L}_{i}$ with respect to the period $A$ up to compatibility.)

    1. (d)

An ‘‘identical’’ case: $\mathsf{M}\sim\mathsf{N}$ and they belong to some
$\mathsf{S}_{i}$ and $\mathsf{S}_{j}$ respectively.

Note that the two operations are reversible and are defined up to
compatibility.

Let $\mathsf{K}$ and $\mathsf{K}^{\prime}$ be fragments in $\mathsf{P}_{0}$
such that
$\mu_{\mathrm{f}}(\mathsf{K}),\mu_{\mathrm{f}}(\mathsf{K}^{\prime})\geq\xi_{1}$
and $\mathsf{K}^{\prime}\sim s_{A,\mathsf{P}_{0}}\mathsf{K}$, as assumed in
the proposition. Let $\mathcal{M}$ be a maximal set of pairwise non-compatible
fragments which can be obtained by operations (i) and (ii) starting from
$\mathsf{K}$. By Proposition 8.10, neither of any two fragments in
$\mathcal{M}$ is contained in the other, so $\mathcal{M}$ is a finite set.

The following assertion is the principal step of the proof.

  * Claim:

The jump operation is always possible inside $\mathcal{M}$; that is, for any
$\mathsf{M}\in\mathcal{M}$ in $\mathsf{P}_{i}$ or in $\mathsf{S}_{i}$,
$i\in\\{0,1\\}$, there exists a fragment $\mathsf{N}$ of rank $\alpha$ in
$\mathsf{S}_{i}$ or, respectively, in $\mathsf{P}_{i}$ such that
$\mathsf{M}\sim\mathsf{N}^{\pm 1}$.

Proof of the claim. We assume that some $\mathsf{M}\in\mathcal{M}$ is given
and prove existence of the required $\mathsf{N}$. The proof will consist of
application of Lemma 13.1. We do a necessary preparation.

According to the definition of $\mathcal{M}$, there is a sequence
$\mathsf{T}_{0}=\mathsf{K}$, $\mathsf{T}_{1}$, $\dots$,
$\mathsf{T}_{l}=\mathsf{M}$ of fragments $\mathsf{T}_{j}\in\mathcal{M}$ such
that $\mathsf{T}_{j+1}$ is obtained from $\mathsf{T}_{j}$ by one of the
operations (i) or (ii). We can assume that the sequence has no two
translations in a row (otherwise we can replace them by a single translation)
and has no two jumps in a row (otherwise they eliminate). Assume also for
convenience that $\mathsf{T}_{0}\to\mathsf{T}_{1}$ is a translation (by
inserting a trivial translation if needed). Thus for each $i$,
$\mathsf{T}_{2j}$ translates to $\mathsf{T}_{2j+1}$ and $\mathsf{T}_{2j+1}$
jumps to $\mathsf{T}_{2j+2}$. We can assume that the last step
$\mathsf{T}_{l-1}\to\mathsf{T}_{l}$ is a translation, so $l=2k-1$ for some
$k$.

Now roughly speaking, we move all fragments $\mathsf{T}_{j}$ along with the
corresponding paths $\mathsf{P}_{i}$ or $\mathsf{S}_{i}$ belonging them, to
the same location up to compatibility. We define a sequence $\mathsf{Y}_{0}$,
$\mathsf{Y}_{1}$, $\dots$, $\mathsf{Y}_{k}$ of paths in $\Gamma_{\alpha}$ and
a sequence $\mathsf{W}_{j}$ of fragments in $\mathsf{Y}_{j}$ for
$j=0,1,\dots,k-1$. For each $j$ we will have
$\mathsf{W}_{j}=f_{j}\mathsf{T}_{2j+1}$ for some $f_{j}\in G_{\alpha}$. The
definition of $\mathsf{Y}_{j}$ and $f_{j}$ goes as follows.

Denote
$(\mathsf{X}_{1},\mathsf{X}_{2},\mathsf{X}_{3},\mathsf{X}_{4})=(\mathsf{P}_{0},\mathsf{S}_{0},\mathsf{P}_{1},\mathsf{S}_{1})$
and let $J(i)$ denote the index such that a fragment in $\mathsf{X}_{i}$ jumps
to a fragment in $\mathsf{X}_{J(i)}$ (i.e. $(J(1),J(2),J(3),J(4))=(2,1,4,3)$).
Denote also $I(j)$ the index such that $\mathsf{T}_{2j-1}$ belongs to
$\mathsf{X}_{I(j)}$. Thus, $\mathsf{T}_{2j}$ belongs to
$\mathsf{X}_{J(I(j))}$.

We start with $\mathsf{Y}_{0}=\mathsf{X}_{I(0)}$ and
$\mathsf{W}_{0}=\mathsf{T}_{1}$, so $f_{0}=1$. Assume that $j<k-1$ and
$\mathsf{Y}_{j}$ and $f_{j}$ are already defined. If
$\mathsf{T}_{2j}\to\mathsf{T}_{2j+1}$ is a translation by (a)–(c) then there
exists $f_{j+1}\in G_{\alpha}$ such that $f_{j+1}\mathsf{X}_{I(j+1)}$ and
$f_{j}\mathsf{X}_{J(I(j))}$ belong to the same $A$-periodic line and
$f_{j+1}\mathsf{T}_{2j+1}\sim f_{j}\mathsf{T}_{2j}$. We take
$\mathsf{Y}_{j+1}=f_{j+1}\mathsf{X}_{I(j+1)}\cup f_{j}\mathsf{X}_{J(I(j))}$.
Otherwise $\mathsf{T}_{2j}\to\mathsf{T}_{2j+1}$ is a translation by (d), i.e.
$\mathsf{X}_{J(I(j))}$ is either $\mathsf{S}_{0}$ or $\mathsf{S}_{1}$. In this
case we take $f_{j+1}=f_{j}$ and $\mathsf{Y}_{j+1}=f_{j}\mathsf{S}_{1}$.
Finally, define $\mathsf{Y}_{k}=f_{k}\mathsf{X}_{J(I(k-1))}$. We have
$f_{j+1}\mathsf{T}_{2j+2}^{\pm 1}\sim f_{j+1}\mathsf{T}_{2j+1}\sim
f_{j}\mathsf{T}_{2j}$ for all $j=0,1,\dots,k-2$ and hence
$\mathsf{W}_{0}\sim\mathsf{W}_{1}^{\pm 1}\sim\dots\sim\mathsf{W}_{k-1}^{\pm
1}$. Figure 37 illustrates the construction.

$\mathsf{W}_{0}$
---
$\mathsf{W}_{1}$
---
$\mathsf{W}_{2}$
---
$\mathsf{W}_{3}$
---
Figure 37.

By strict closeness of pairs $(\mathsf{P}_{0},\mathsf{S}_{0})$ and
$(\mathsf{P}_{1},\mathsf{S}_{1})$, each $\mathsf{X}_{i}$ starts with a
fragment $\mathsf{U}_{i}$ and ends with a fragment $\mathsf{V}_{i}$ such that
$\mu_{\mathrm{f}}(\mathsf{U}_{i}),\mu_{\mathrm{f}}(\mathsf{V}_{i})\geq\xi_{2}$,
$\mathsf{U}_{i}\not\sim\mathsf{V}_{i}$ and we have
$\mathsf{U}_{i}\sim\mathsf{U}_{J(i)}^{\pm 1}$ and
$\mathsf{V}_{i}\sim\mathsf{V}_{J(i)}^{\pm 1}$

We now apply Lemma 13.1 where:

  * •

$S_{j}$ is the set of all fragments $\mathsf{N}$ in $\mathsf{S}_{j}$ with
$\mu_{\mathrm{f}}(\mathsf{N})\geq\xi_{3}$.

  * •

$\mathsf{N}<_{i}\mathsf{N}^{\prime}$ is defined as
‘$\mathsf{N}\not\sim\mathsf{N}^{\prime}$ and $\mathsf{N}<\mathsf{N}^{\prime}$
in $\mathsf{S}_{j}$’.

  * •

Equivalence of $\mathsf{N},\mathsf{N}^{\prime}\in\bigcup_{j}S_{j}$ is defined
as $\mathsf{N}\sim\mathsf{N}^{\prime\pm 1}$.

  * •

$\mathsf{N}\in\bigcup_{j}S_{j}$ is defined to be stable iff
$\mu_{\mathrm{f}}(\mathsf{N})\geq\xi_{2}$.

  * •

For $a_{j}$, $b_{j}$, $a_{j}^{\prime}$ and $b_{j}^{\prime}$ we take
appropriate translates of $\mathsf{U}_{i}$ and $\mathsf{V}_{i}$, namely,
$f_{j}\mathsf{U}_{I(j)}$, $f_{j}\mathsf{V}_{I(j)}$,
$f_{j}\mathsf{U}_{J(I(j))}$ and $f_{j}\mathsf{V}_{J(I(j))}$ respectively.

We have conditions (i)–(v) of Lemma 13.1 satisfied: condition (i) holds in
case $\beta\geq 1$ by Corollary 9.24(ii), condition (ii) holds by Proposition
8.10β, conditions (iii) and (iv) hold by Proposition 10.23 in view of the
inequality $\xi_{3}\geq 2\lambda+9.1\omega$ and, finally, condition (v) holds
immediately by construction.

For $c_{0}$ in Lemma 13.1 we take $\mathsf{T}_{1}$. Note that up to
compatibility, we can assume that
$\mu_{\mathrm{f}}(\mathsf{T}_{1})\geq\xi_{2}$, so $\mathsf{T}_{1}$ is stable.
(By construction, $\mathsf{T}_{1}$ is obtained from
$\mathsf{T}_{0}=\mathsf{K}$ by translation to $\mathsf{X}_{I(0)}$; if
$\mathsf{T}_{1}$ is compatible with the starting or the ending fragment of
$\mathsf{X}_{I(0)}$ then we can assume
$\mu_{\mathrm{f}}(\mathsf{T}_{1})\geq\xi_{2}$ due to Definition 13.3;
otherwise we can assume that $\mathsf{T}_{1}$ is a literal translation of
$\mathsf{K}$ and then
$\mu_{\mathrm{f}}(\mathsf{T}_{1})=\mu_{\mathrm{f}}(\mathsf{K})\geq\xi_{1}$.)
Since $\mathsf{T}_{1}=\mathsf{W}_{0}\sim\mathsf{W}_{1}^{\pm
1}\sim\dots\sim\mathsf{W}_{k-1}^{\pm 1}$ and each $\mathsf{W}_{j}$ occurs in
$f_{j}\mathsf{X}_{I(j)}$, $\mathsf{T}_{1}$ penetrates between each pair
$f_{j}\mathsf{U}_{I(j)}$ and $f_{j}\mathsf{V}_{I(j)}$ for $j=0,1,\dots,k-1$.
All the hypotheses of Lemma 13.1 are satisfied and applying it we find a
fragment $\mathsf{W}_{k}$ in $f_{k-1}\mathsf{X}_{J(I(k-1))}$ such that
$\mathsf{W}_{k}^{\pm 1}\sim\mathsf{W}_{k-1}=f_{k-1}\mathsf{M}$. Then
$\mathsf{M}\to f_{k-1}^{-1}\mathsf{W}_{k}$ is the required jump. This finishes
the proof of the claim.

We finish the proof of the proposition. Let $\mathsf{K}_{0}=\mathsf{K}$,
$\mathsf{K}_{1}$, $\dots$, $\mathsf{K}_{m}\sim s_{A,\mathsf{P}_{0}}\mathsf{K}$
be all fragments in $\mathcal{M}$ between $\mathsf{K}$ and
$s_{A,\mathsf{P}_{0}}\mathsf{K}$ in their natural order, i.e. we have
$\mathsf{K}_{0}<\mathsf{K}_{1}<\dots<\mathsf{K}_{m}$. Let
$\mathsf{M}_{0},\dots,\mathsf{M}_{m}\in\mathcal{M}$ be fragments in
$\mathsf{P}_{1}$ such that $\mathsf{M}_{i}\sim\mathsf{K}_{i}^{\pm 1}$ for all
$i$ (each $\mathsf{M}_{i}$ is obtained from $\mathsf{K}_{i}$ by two jumps).
Note that $\mathsf{M}_{0}<\mathsf{M}_{1}<\dots<\mathsf{M}_{m}$ by Proposition
10.23. Since $\mathcal{M}$ is closed under translations, the number of
fragments in $\mathcal{M}$ between $\mathsf{M}_{0}$ and
$s_{A,\mathsf{P}_{1}}\mathsf{M}_{0}$ is the same as the number of fragments in
$\mathcal{M}$ between $\mathsf{K}$ and $s_{A,\mathsf{P}_{0}}\mathsf{K}$, i.e.
we have $\mathsf{M}_{m}\sim s_{A,\mathsf{P}_{1}}\mathsf{M}_{0}$. This implies
that $\mathsf{K}_{0}$ translates to some $\mathsf{M}_{q}$, i.e.
$\mathsf{M}_{q}\sim gs_{A,\mathsf{P}_{0}}^{t}\mathsf{K}_{0}^{\pm 1}$ for some
$t$ and hence

$\mathsf{M}_{i+q}\sim gs_{A,\mathsf{P}_{0}}^{t}\mathsf{K}_{i}^{\pm 1}\
\text{for}\ i=0,1,\dots,m-q,\quad\mathsf{M}_{i+q-m}\sim
gs_{A,\mathsf{P}_{0}}^{t-1}\mathsf{K}_{i}^{\pm 1}\ \text{for}\
i=m-q+1,\dots,m.$

Note that $\gcd(m.q)=1$ since $\mathcal{M}$ is generated by a single fragment
$\mathsf{K}$. By Propositions 8.16(i), 11.12 and Corollary 9.24(iii), the
subgroup $\\{g\in G_{\alpha}\;|\;g\mathsf{M}_{0}\sim\mathsf{M}_{0}^{\pm 1}\\}$
is malnormal in $G_{\alpha}$. We now apply Lemma 13.2 where for $x_{i}$ we
take the equivalence class of $\mathsf{M}_{i}$ in the set of fragments of rank
$\beta$ in $\Gamma_{\alpha}$ under compatibility up to invertion. By the
lemma, $\langle g,s_{A,\mathsf{P}_{0}}\rangle$ is cyclic. Since $A$ is a non-
power, we get $g\in\langle s_{A,\mathsf{P}_{0}}\rangle$ which means that
$\mathsf{L}_{1}=\mathsf{L}_{2}$. ∎

As an immediate consequence of Proposition 13.4 we get:

######  13.5 Corollary (overlapping coarse periodicity).

Let $\mathsf{S}_{0}$ and $\mathsf{S}_{1}$ be coarsely periodic segments in
$\Gamma_{\alpha}$ with the same simple period $A$ over $G_{\alpha}$. If
$\mathsf{S}_{0}$ is contained in $\mathsf{S}_{1}$ then
$\mathsf{S}_{0}\sim\mathsf{S}_{1}$.

######  13.6 Corollary.

Let $\mathsf{S}$ and $\mathsf{T}$ be non-compatible coarse periodic segments
in $\Gamma_{\alpha}$ with the same simple period $A$ which occur in a reduced
path $\mathsf{X}$. Let $\ell_{A}(\mathsf{S})\geq 3$. Assume that
$\mathsf{S}_{1}$ is obtained from $\mathsf{S}$ by shortening by 2 periods from
the end if $\mathsf{S}<\mathsf{T}$ or by shortening by 2 periods from the
start if $\mathsf{S}>\mathsf{T}$. Then $\mathsf{S}_{1}$ and $\mathsf{T}$ are
disjoint.

###### Proof.

Without loss of generality, we assume that $\mathsf{S}<\mathsf{T}$ and
$\mathsf{S}_{1}$ is obtained from $\mathsf{S}$ by shortening by 2 periods from
the end. By 12.10(iii) we have
$\mathsf{S}=\mathsf{S}_{1}\mathsf{u}\mathsf{S}_{2}$ where $\mathsf{S}_{2}$ is
a coarsely $A$-periodic segment with $\mathsf{S}_{2}\sim\mathsf{S}$. B y
hypothesis we have $\mathsf{S}_{2}\not\sim\mathsf{T}$ and then by Corollary
13.5, neither of $\mathsf{S}_{2}$ or $\mathsf{T}$ is contained in the other.
This implies that $\mathsf{S}_{1}$ and $\mathsf{T}$ are disjoint. ∎

######  13.7 Proposition (strictly close periodic paths with one period).

Let $A$ be a simple period over $G_{\alpha}$ and $\beta$ the activity rank of
$A$. Let $\mathsf{P}_{0}$ and $\mathsf{P}_{1}$ be strictly close in rank
$\beta$ paths in $\Gamma_{\alpha}$ labeled by periodic words with period $A$.
Assume that there exist fragments $\mathsf{K},\mathsf{K}^{\prime}$ of rank
$\beta$ in $\mathsf{P}_{0}$ such that
$\mu_{\mathrm{f}}(\mathsf{K}),\mu_{\mathrm{f}}(\mathsf{K}^{\prime})\geq\xi_{2}$
and $s_{A,\mathsf{P}_{0}}\mathsf{K}\sim\mathsf{K}^{\prime}$. Then
$\mathsf{P}_{0}$ and $\mathsf{P}_{1}$ have a common periodic extension.

###### Proof.

This is a special case of Proposition 13.4 with
$\mathsf{S}_{0}=\mathsf{S}_{1}=\mathsf{P}_{1}$. ∎

######  13.8 Proposition.

Let $g\in G_{\alpha}$ be a non-power of infinite order and let $h\in
G_{\alpha}$. If $g^{k}=h^{-1}g^{l}h$ for some $k,l>0$ then $h\in\langle
g\rangle$ and $k=l$.

###### Proof.

By Proposition 11.5, up to conjugation we can assume that $g$ is represented
by a simple period $A$ over $G_{\alpha}$. It is enough to prove that
$h\in\langle A\rangle$.

Consider two periodic lines $\mathsf{L}_{0}$ and $\mathsf{L}_{1}$ in
$G_{\alpha}$ with period $A$ which represent the conjugacy relation. We have
$h\in\langle A\rangle$ if and only if $\mathsf{L}_{0}=\mathsf{L}_{1}$. Let
$\beta$ be the activity rank of $A$. By Proposition 10.24 we find strictly
close in rank $\beta$ subpaths $\mathsf{P}_{i}$ of $\mathsf{L}_{i}$ with any
desired bound $|\mathsf{P}_{0}|\geq t|A|$. Then the statement follows from
Proposition 13.7. ∎

As an immediate consequence we get:

######  13.9 Corollary.

Let $\mathsf{S}_{0}$ and $\mathsf{S}_{1}$ be coarsely $A$-periodic segments in
$\Gamma_{\alpha}$ and $\mathsf{L}_{i}$ $(i=1,2)$ be an axis for
$\mathsf{S}_{i}$. If $\mathsf{S}_{0}\sim\mathsf{S}_{1}$ then
$\mathsf{L}_{1}=\mathsf{L}_{2}$.

######  13.10 Corollary.

Let $g\in G_{\alpha}$ be an element of infinite order. Then the following is
true.

  1. (i)

$g$ has the unique root; i.e. there exists a unique non-power element
$g_{0}\in G_{\alpha}$ such that $g=g_{0}^{t}$ for some $t\geq 1$.

  2. (ii)

If $h^{r}\in\langle g\rangle$ and $h^{r}\neq 1$ then $h\in\langle
g_{0}\rangle$ where $g_{0}$ is the root of $g$.

  3. (iii)

If $g$ is conjugate to $g^{-1}$ then $g$ is the product of two involutions.

###### Proof.

(i) is direct consequence of Propositions 11.13 and 13.8.

(ii) follows from (i) and Proposition 13.8 because $g_{0}^{t}=h^{r}$ implies
$g_{0}^{t}=h^{-1}g_{0}^{t}h$.

(iii) Assume that $g=h^{-1}g^{-1}h$. From $g=h^{-2}gh^{2}$ we conclude that
$h^{2}=1$ by (ii). Similarly, we have $(hg)^{2}=1$ and then $g=h\cdot hg$. ∎

######  13.11 Corollary.

Assume that each relator $R$ of each rank $\beta\leq\alpha$ has the form
$R=R_{0}^{n}$ where $R_{0}$ is the root of $R$ and $n$ is odd ($n$ can vary
for different relators $R$). Then $G_{\alpha}$ has no involutions and no
element of $G_{\alpha}$ is conjugate to its inverse.

###### Proof.

By Proposition 11.5, any element of finite order of $G_{\alpha}$ is conjugate
to some power $R_{0}^{t}$ of the root $R_{0}$ of a relator $R$ of rank
$\beta\leq\alpha$. By Proposition 11.11, $R_{0}^{t}$ has an odd order and
cannot be an involution. The second statement follows from the first by
Corollary 13.10(iii). ∎

######  13.12 Lemma.

Let $\mathsf{P}$ be an $A$-periodic segment in $\Gamma_{\alpha}$ with a simple
period $A$ over $G_{\alpha}$. Let $\mathsf{S}$ be a coarsely periodic segment
in $\mathsf{P}$ with another simple period $B$ over $G_{\alpha}$ and assume
that $A$ and $B$ are not conjugate in $G_{\alpha}$. Then the following is
true.

  1. (i)

$\mathsf{S}\not\sim s_{A,\mathsf{P}}^{t}\mathsf{S}$ for any $t\neq 0$.

  2. (ii)

If $\ell_{B}(\mathsf{S})\geq 3$ then $|\mathsf{S}|<2|A|$.

###### Proof.

(i) Assume that $\mathsf{S}\sim s_{A,\mathsf{P}}^{t}\mathsf{S}$ for some
$t\neq 0$. Let $\mathsf{L}_{1}$ be the infinite periodic extension of
$\mathsf{P}$, and let $\mathsf{L}_{2}$ be the axis for $\mathsf{K}$. By
Corollary 13.9 we have $\mathsf{L}_{2}=s_{A,\mathsf{P}}^{t}\mathsf{L}_{2}$, so
$s_{A,\mathsf{P}}^{t}=s_{B,\mathsf{Q}}^{r}$ for some $r\neq 0$. Since $A$ and
$B$ are non-powers, by Corollary 13.10(ii)
$s_{A,\mathsf{P}}^{\varepsilon}=s_{B,\mathsf{Q}}$ for $\varepsilon=\pm 1$ and
hence $L_{1}^{\varepsilon}$ and $L_{2}$ are parallel. From the fact that
$\mathsf{S}$ is a subpath of $\mathsf{P}$ we easily deduce by Proposition
10.23 (taking for $\beta$ the activity rank of $A$) that $\varepsilon=1$. We
obtain a contradiction with the assumption that $A$ and $B$ are not conjugate
in $G_{\alpha}$.

(ii) By 12.10(iii) we represent $\mathsf{S}$ as
$\mathsf{S}=\mathsf{S}_{1}\mathsf{u}\mathsf{S}_{2}$ where $\mathsf{S}_{1}$ and
$\mathsf{S}_{2}$ are coarsely periodic segments with period $B$ and
$\ell_{B}(\mathsf{S}_{1})\geq\ell_{B}(\mathsf{S})-2$. By (i) and Corollary
13.5, $s_{A,\mathsf{P}}^{-1}\mathsf{S}$ does not contain $\mathsf{S}_{1}$ and
$s_{A,\mathsf{P}}\mathsf{S}$ does not contain $\mathsf{S}_{2}$. This implies
$|\mathsf{S}|<2|A|$. ∎

######  13.13 Proposition.

Let $\mathsf{P}$ and $\mathsf{Q}$ be close periodic segments in
$\Gamma_{\alpha}$ with the same simple period $A$ over $G_{\alpha}$. If
$|\mathsf{P}|\geq(2h_{\alpha}(A)+1)|A|$ (where $h_{\alpha}(A)$ is defined in
12.12) then $\mathsf{P}$ and $\mathsf{Q}$ belong to the same $A$-periodic
line.

###### Proof.

Follows from Propositions 12.15 and 13.7. ∎

We finish the section by formulating technical statements which we will need
in the construction of relations of Burnside groups. We use notation
$\mathsf{S}\lessapprox\mathsf{T}$ for ‘$\mathsf{S}<\mathsf{T}$ or
$\mathsf{S}\approx\mathsf{T}$’.

######  13.14 Lemma.

Let $\mathsf{S}$ and $\mathsf{T}$ be coarsely $A$-periodic segments occurring
in a reduced path $\mathsf{X}$ in $\Gamma_{\alpha}$. Assume that some periodic
bases for $\mathsf{S}$ and $\mathsf{T}$ have the same label. If $\mathsf{S}$
is contained in $\mathsf{T}$ then $\mathsf{S}\approx\mathsf{T}$.

###### Proof.

Assume that $\mathsf{S}$ is contained in $\mathsf{T}$. Let $\mathsf{P}_{i}$
$(i=1,2)$ be periodic bases for $\mathsf{S}$ and $\mathsf{T}$ respectively,
with
$\operatorname{\mathit{label}}(\mathsf{P}_{1})\eqcirc\operatorname{\mathit{label}}(\mathsf{P}_{2})$.
Let $\beta$ be the activity rank of $A$. By Proposition 13.4, $\mathsf{P}_{1}$
and $\mathsf{P}_{2}$ have a common periodic extension. Let $\mathsf{K}_{i}$
and $\mathsf{M}_{i}$ $(i=0,1,2,3)$ be fragments of rank $\beta$ with
$\mu_{\mathrm{f}}(\mathsf{K}_{i}),\mu_{\mathrm{f}}(\mathsf{M}_{i})\geq\xi_{2}$
such that $\mathsf{P}_{1}=\mathsf{K}_{0}\cup\mathsf{K}_{1}$,
$\mathsf{P}_{2}=\mathsf{K}_{2}\cup\mathsf{K}_{3}$,
$\mathsf{S}=\mathsf{M}_{0}\cup\mathsf{M}_{1}$,
$\mathsf{T}=\mathsf{M}_{2}\cup\mathsf{M}_{3}$ and
$\mathsf{K}_{i}\sim\mathsf{M}_{i}$ for all $i$. We have
$\mathsf{M}_{2}\lesssim\mathsf{M}_{0}\lnsim\mathsf{M}_{1}\lesssim\mathsf{M}_{3}$
which by Proposition 13.4 implies $\mathsf{K}_{2}\lesssim\mathsf{K}_{0}$ and
$\mathsf{K}_{1}\lesssim\mathsf{K}_{3}$. Now from
$\operatorname{\mathit{label}}(\mathsf{P}_{1})\eqcirc\operatorname{\mathit{label}}(\mathsf{P}_{2})$
we conclude that $\mathsf{K}_{2}\sim\mathsf{K}_{0}$ and
$\mathsf{K}_{1}\sim\mathsf{K}_{3}$, i.e. $\mathsf{S}\approx\mathsf{T}$. ∎

######  13.15 Lemma.

Let $\mathsf{X}$ and $\mathsf{Y}$ be close reduced paths in $\Gamma_{\alpha}$.
Let $\mathsf{S}_{0},\mathsf{S}_{1}$ be coarsely $A$-periodic segments in
$\mathsf{X}$ and $\mathsf{T}_{0},\mathsf{T}_{1}$ be coarsely $A$-periodic
segments in $\mathsf{Y}$ such that $\ell(\mathsf{S}_{i})\geq
2h_{\alpha}(A)+1$, $\mathsf{S}_{i}\approx\mathsf{T}_{i}$ for $i=0,1$ and
$\mathsf{S}_{0}\not\sim\mathsf{S}_{1}$. Then $\mathsf{S}_{0}<\mathsf{S}_{1}$
if and only if $\mathsf{T}_{0}<\mathsf{T}_{1}$.

###### Proof.

By Corollary 13.5, none of $\mathsf{S}_{0}$ and $\mathsf{S}_{1}$ is contained
in the other and the same is true for $\mathsf{T}_{0}$ and $\mathsf{T}_{1}$.
Assume, for example, that $\mathsf{S}_{0}<\mathsf{S}_{1}$ and
$\mathsf{T}_{1}<\mathsf{T}_{0}$. Let $\mathsf{X}_{1}$ and $\mathsf{Y}_{1}$ be
the starting segments of $\mathsf{X}$ and $\mathsf{Y}$ ending with
$\mathsf{S}_{1}$ and $\mathsf{S}_{2}$ respectively. By Proposition 12.14 with
$\mathsf{X}:=\mathsf{X}_{1}$ and $\mathsf{Y}:=\mathsf{Y}_{1}$ there exists
$\mathsf{U}$ in $Y_{1}$ such that $\mathsf{U}\approx\mathsf{S}_{0}^{*}$ where
$\mathsf{S}_{0}^{*}$ is the stable part of $\mathsf{S}_{0}$. Then
$\mathsf{U}\cup\mathsf{T}_{0}$ is a coarsely $A$-periodic segment containing
$\mathsf{T}_{1}$ and we get a contradiction with Corollary 13.5. ∎

######  13.16 Lemma.

Let $\mathsf{X}$ and $\mathsf{Y}$ be reduced paths in $\Gamma_{\alpha}$. Let
$\mathsf{S}_{0},\mathsf{S}_{1}$ be coarsely $A$-periodic segments in
$\mathsf{X}$ and $\mathsf{T}_{0},\mathsf{T}_{1}$ be coarsely $A$-periodic
segments in $\mathsf{Y}$ such that $\mathsf{S}_{0}\lessapprox\mathsf{S}_{1}$,
$\mathsf{T}_{0}\lessapprox\mathsf{T}_{1}$ and
$\mathsf{S}_{i}\approx\mathsf{T}_{i}$, $i=0,1$.

  1. (i)

Let $\mathsf{U}$ be a coarsely $A$-periodic segment in $\mathsf{X}$ such that
$\mathsf{S}_{0}\lessapprox\mathsf{U}\lessapprox\mathsf{S}_{1}$,
$\ell_{A}(\mathsf{U})\geq h_{\alpha}(A)+1$ and $\mathsf{U}$ is the stable part
of some other coarsely $A$-periodic segment in $\mathsf{X}$. Then there exists
a coarsely $A$-periodic segment $\mathsf{V}$ in $\mathsf{Y}$ such that
$\mathsf{T}_{0}\lessapprox\mathsf{V}\lessapprox\mathsf{T}_{1}$ and
$\mathsf{U}\approx\mathsf{V}$.

  2. (ii)

Let $\mathsf{U}_{i}$ $(i=1,2)$ be coarsely $A$-periodic segments in
$\mathsf{X}$ and $\mathsf{V}_{i}$ $(i=1,2)$ be coarsely $A$-periodic segments
in $\mathsf{Y}$ such that $\ell_{A}(\mathsf{U}_{i})\geq 2h_{\alpha}(A)+1$
$(i=1,2)$, $\mathsf{S}_{0}\lessapprox\mathsf{U}_{i}\lessapprox\mathsf{S}_{1}$,
$\mathsf{T}_{0}\lessapprox\mathsf{V}_{i}\lessapprox\mathsf{T}_{1}$ and
$\mathsf{U}_{i}\approx\mathsf{V}_{i}$ for $i=1,2$. Assume that
$\mathsf{U}_{2}\approx g\mathsf{U}_{1}$ for some $g\in G_{\alpha}$, i.e.
$\mathsf{U}_{1}$ and $\mathsf{U}_{2}$ have periodic bases with the same label.
Then $\mathsf{U}_{1}\lessapprox\mathsf{U}_{2}$ if and only if
$\mathsf{V}_{1}\lessapprox\mathsf{V}_{2}$.

###### Proof.

Let $\beta$ be the activity rank of $A$.

(i): Let $\mathsf{U}$ be the stable part of $\bar{\mathsf{U}}$ and
$\bar{\mathsf{U}}=\mathsf{Z}_{1}\mathsf{U}\mathsf{Z}_{2}$. We consider several
cases.

Case 1: $\mathsf{U}\not\sim\mathsf{S}_{i}$ for $i=0,1$. Then by Corollary 13.5
we have $\mathsf{S}_{0}<\bar{\mathsf{U}}<\mathsf{S}_{1}$. Since
$\mathsf{S}_{0}\cup\mathsf{S}_{1}$ and $\mathsf{T}_{0}\cup\mathsf{T}_{1}$ are
close, existence of $\mathsf{V}$ follows from Proposition 12.14.

Case 2: Exactly one of the relations $\mathsf{U}\sim\mathsf{S}_{i}$ $(i=0,1)$
holds. Without loss of generality, assume that $\mathsf{U}\sim\mathsf{S}_{0}$
and $\mathsf{U}\not\sim\mathsf{S}_{1}$. By Corollary 13.5 we have
$\bar{\mathsf{U}}<\mathsf{S}_{1}$. If $\mathsf{U}\approx\mathsf{S}_{0}$ there
is nothing to prove. Assume that $\mathsf{U}\not\approx\mathsf{S}_{0}$ and
hence $\mathsf{U}\mathsf{Z}_{2}$ is contained in
$\mathsf{S}_{0}\cup\mathsf{S}_{1}$.

By the construction of the stable part, $\mathsf{U}\mathsf{Z}_{2}$ is a
coarsely $A$-periodic segment with
$\ell_{A}(\mathsf{U}\mathsf{Z}_{2})=\ell(\mathsf{U})+h_{\alpha}(A)\geq
2h_{\alpha}(A)+1$. Let $\mathsf{W}$ be the stable part of
$\mathsf{U}\mathsf{Z}_{2}$. Using Proposition 12.14 with
$\mathsf{X}:=\mathsf{S}_{0}\cup\mathsf{S}_{1}$ and
$\mathsf{Y}:=\mathsf{T}_{0}\cup\mathsf{T}_{1}$ we find a coarsely $A$-periodic
segment $\mathsf{W}^{\prime}$ in $\mathsf{T}_{0}\cup\mathsf{T}_{1}$ such that
$\mathsf{W}\approx\mathsf{W}^{\prime}$. By Proposition 12.9(ii),

$\mathsf{S}_{0}\cup\mathsf{U}$ is a coarsely $A$-periodic segment and since
$\mathsf{W}^{\prime}\sim\mathsf{T}_{0}$,
$\mathsf{T}_{0}\cup\mathsf{W}^{\prime}$ is a coarsely $A$-periodic segment as
well. By 12.10(iv) (more formally, by the symmetric version of 12.10(iv))
$\mathsf{W}$ is an end of $\mathsf{U}$ which implies
$\mathsf{S}_{0}\cup\mathsf{U}\approx\mathsf{T}_{0}\cup\mathsf{W}^{\prime}$.

Now let $\mathsf{P}$ be a periodic base for $\mathsf{U}$. By the construction
of the stable part, $\mathsf{P}$ starts with a fragment $\mathsf{N}$ of rank
$\beta$ with $\mu_{\mathrm{f}}(\mathsf{N})\geq\xi_{1}$. Since $\mathsf{P}$ is
contained in a periodic base for $\mathsf{T}_{0}\cup\mathsf{W}^{\prime}$, by
Proposition 10.23 we find a fragment $\mathsf{N}^{\prime}$ of rank $\beta$ in
$\mathsf{T}_{0}\cup\mathsf{W}^{\prime}$ such that
$\mu_{\mathrm{f}}(\mathsf{N}^{\prime})\geq\xi_{2}$ and
$\mathsf{N}^{\prime}\sim\mathsf{N}$. Then for the desired $\mathsf{V}$ we can
take the end of $\mathsf{T}_{0}\cup\mathsf{W}^{\prime}$ starting with
$\mathsf{N}^{\prime}$.

Case 3: $\mathsf{U}\sim\mathsf{S}_{0}\sim\mathsf{S}_{1}$. Then a periodic base
$\mathsf{P}$ for $\mathsf{U}$ is contained in a periodic base for
$\mathsf{S}_{0}\cup\mathsf{S}_{1}$. By the construction of the stable part,
$\mathsf{P}$ starts and ends with fragments $\mathsf{N}_{0}$ and
$\mathsf{N}_{1}$ of rank $\beta$ with
$\mu_{\mathrm{f}}(\mathsf{N}_{i})\geq\xi_{1}$. Then using Proposition 10.23 we
find fragments $\mathsf{N}_{i}^{\prime}$ $(i=0,1)$ of rank $\beta$ in
$\mathsf{T}_{0}\cup\mathsf{T}_{1}$ such that
$\mu_{\mathrm{f}}(\mathsf{N}_{i}^{\prime})\geq\xi_{2}$ and
$\mathsf{N}_{i}^{\prime}\sim\mathsf{N}_{i}$ $(i=1,2)$. We can take
$\mathsf{V}=\mathsf{N}_{0}^{\prime}\cup\mathsf{N}_{1}^{\prime}$.

(ii): We consider two cases.

Case 1: $\mathsf{U}_{1}\sim\mathsf{U}_{2}$. Let $\mathsf{P}_{1}$ and
$\mathsf{P}_{2}$ be periodic bases for $\mathsf{U}_{1}$ and $\mathsf{U}_{2}$
with
$\operatorname{\mathit{label}}(\mathsf{P}_{1})\eqcirc\operatorname{\mathit{label}}(\mathsf{P}_{2})$
which have a common periodic extension. It easily follows from Proposition
10.23(ii) that
$\mathsf{U}_{1}<\mathsf{U}_{2}\Leftrightarrow\mathsf{P}_{1}<\mathsf{P}_{2}$
and
$\mathsf{U}_{1}\approx\mathsf{U}_{2}\Leftrightarrow\mathsf{P}_{1}=\mathsf{P}_{2}$.
Since $\mathsf{P}_{i}$ is also a periodic base for $\mathsf{V}_{i}$, a similar
statement holds for $\mathsf{V}_{i}$’s which clearly implies the required
conclusion.

Case 2: $\mathsf{U}_{1}\not\sim\mathsf{U}_{2}$. Without loss of generality, we
assume that $\mathsf{U}_{1}<\mathsf{U}_{2}$, $\mathsf{V}_{1}>\mathsf{V}_{2}$
and come to a contradiction. We can assume also that
$\mathsf{X}=\mathsf{S}_{0}\cup\mathsf{S}_{1}$,
$\mathsf{Y}=\mathsf{T}_{0}\cup\mathsf{T}_{1}$ and hence $\mathsf{X}$ and
$\mathsf{Y}$ are close in rank $\alpha$. Let $\mathsf{U}_{i}^{*}$ and
$\mathsf{V}_{i}^{*}$ be stable parts of $\mathsf{U}_{i}$ and $\mathsf{V}_{i}$.
By Corollary 13.6, $\mathsf{U}_{1}$ is disjoint from $\mathsf{U}_{2}^{*}$. Let
$\mathsf{X}=\mathsf{X}_{1}\mathsf{U}_{1}\mathsf{X}_{2}\mathsf{U}_{2}^{*}\mathsf{X}_{3}$
and
$\mathsf{Y}=\mathsf{Y}_{1}\mathsf{V}_{2}^{*}\mathsf{Y}_{2}\mathsf{V}_{1}\mathsf{Y}_{3}$.
By Proposition 12.14 with
$\mathsf{X}=\mathsf{X}_{1}\mathsf{U}_{1}\mathsf{X}_{2}$ and
$\mathsf{Y}:=\mathsf{Y}_{1}$ there exists a coarsely $A$-periodic segment
$\mathsf{W}$ in $\mathsf{Y}_{1}$ such that
$\mathsf{W}\approx\mathsf{U}_{1}^{*}$. Then
$\mathsf{W}\sim\mathsf{U}_{1}\sim\mathsf{V}_{1}$ and by Proposition 12.9(ii)
and Corollary 13.5 we get
$\mathsf{U}_{1}\sim\mathsf{W}\sim\mathsf{W}\cup\mathsf{V}_{1}\sim\mathsf{V}_{2}\sim\mathsf{U}_{2}$,
the desired contradiction. ∎

##  14\. Comparing $\alpha$-length of close words

In this section, we prove the following proposition.

######  14.1 Proposition.

Let $X,Y\in\mathcal{R}_{\alpha}$ be close in rank $\alpha$. Then

$|Y|_{\alpha}<1.3|X|_{\alpha}+2.2.$

Recall that a fragment word $F$ of rank $\alpha$ is considered with fixed
associated words $S$, $u$, $v$ and a relator $R$ of rank $\alpha$ such that
$F=uSv$ in $G_{\alpha-1}$, $u,v\in\mathcal{H}_{\alpha-1}$ and $S$ is a subword
of $R^{k}$ for some $k>0$. If $\mathsf{F}$ is a path in $\Gamma_{\alpha-1}$
labeled $F$ then this uniquely defines the base $\mathsf{S}$ for $\mathsf{F}$.

Let $F$ and $G$ be fragments of rank $\alpha$ in a word $X$. Let $\mathsf{X}$
be a path in $\Gamma_{\alpha-1}$ labeled $X$ and $\mathsf{F}$, $\mathsf{G}$
the corresponding subpaths of $\mathsf{X}$. We write $F\sim G$ if
$\mathsf{F}\sim\mathsf{G}$ (so the relation is formally defined for the
occurrences of $F$ and $G$ in $X$).

Recall that the size $|X|_{\alpha}$ of a word $X$ in rank $\alpha$ is the
minimal possible value of $\text{weight}_{\alpha}(\mathcal{F})$ of a
fragmentation $\mathcal{F}$ of rank $\alpha$ of $X$. A fragmentation
$\mathcal{F}$ of rank $\alpha$ of $X$ is a partition $X\eqcirc F_{1}\cdot
F_{2}\cdots F_{k}$ where $F_{i}$ is a nonempty subword of a fragment of rank
$\beta\leq\alpha$. Assuming that each $F_{i}$ is assigned a unique value of
$\beta$, the weight in rank $\alpha$ of $\mathcal{F}$ is defined by formula

$\text{weight}_{\alpha}(\mathcal{F})=m_{\alpha}+\zeta
m_{\alpha-1}+\zeta^{2}m_{\alpha-2}+\dots+\zeta^{\alpha}m_{0}$

where $m_{\beta}$ is the number of subwords of fragments of rank $\beta$ in
$\mathcal{F}$.

We call a fragmentation $\mathcal{F}$ of $X$ minimal if
$\text{weight}_{\alpha}(\mathcal{F})=|X|_{\alpha}$.

We call a subword $F$ of a fragment of rank $\beta\geq 1$ a truncated fragment
of rank $\beta$. We will be assuming that with a truncated fragment $F$ of
rank $\alpha$ there is an associated genuine fragment $\bar{F}$ of rank
$\beta$ such that $F$ is a subword of $\bar{F}$. If $\mathsf{F}$ is a path in
$\Gamma_{\alpha}$ with $\operatorname{\mathit{label}}(\mathsf{F})\eqcirc F$
then we have the associated fragment $\bar{\mathsf{F}}$ in $\Gamma_{\alpha}$
such that $\mathsf{F}$ is a subpath of $\bar{\mathsf{F}}$. Note a truncated
fragment of rank 1 is simply a fragment of rank 1.

We extend the compatibility relation to truncated fragments of rank $\beta$ in
a word $X$ in the following natural way. If $F$ and $G$ are truncated
fragments of rank $\beta$ in $X$ and $\bar{F}$ and $\bar{G}$ their associated
fragments of rank $\beta$ in $\Gamma_{\alpha}$ then $F\sim G$ if and only if
$\bar{F}\sim\bar{G}$.

### 14.2.

Let $\mathcal{F}=F_{1}\cdot F_{2}\cdot\ldots\cdot F_{k}$ be a fragmentation of
rank $\alpha$ of a word $X$. Let $F_{i}$ be a truncated fragment of rank
$\beta\geq 1$ in $\mathcal{F}$. Assume that $F_{i}$ can be extended in $X$ to
a larger truncated fragment $G$ of rank $\beta$, i.e.

$X\eqcirc F_{1}F_{2}\dots F_{p}^{\prime}F_{p}^{\prime\prime}\dots F_{i}\dots
F_{q}^{\prime}F_{q}^{\prime\prime}\dots F_{k}$

where $F_{p}\eqcirc F_{p}^{\prime}F_{p}^{\prime\prime}$, $F_{q}\eqcirc
F_{q}^{\prime}F_{q}^{\prime\prime}$ and $G\eqcirc F_{p}^{\prime\prime}\dots
F_{i}\dots F_{q}^{\prime}$ (here we consider the case $1<i<k$; cases $i=1$ and
$i=k$ differ only in notation). Then we can produce a new fragmentation
$\mathcal{F}^{\prime}$ of rank $\alpha$, $X\eqcirc F_{1}\cdots
F_{p-1}\cdot[F_{p}^{\prime}]\cdot G\cdot[F_{q}^{\prime\prime}]\cdot
F_{q+1}\cdots F_{k}$ where square brackets mean that $F_{p}^{\prime}$ and
$F_{q}^{\prime\prime}$ are absent if empty. We say that $\mathcal{F}^{\prime}$
is obtained from $\mathcal{F}$ by extending $F_{i}$ to $G$. Note that if
$\mathcal{F}$ is minimal then in the case $i>1$, we necessarily have $p=i-1$
and nonempty $F_{p}^{\prime}$ and in the case $i<k$ we necesarily have $q=i+1$
and nonempty $F_{q}^{\prime\prime}$.

######  14.3 Lemma.

Let $\mathcal{F}=F_{1}\cdot F_{2}\cdot\ldots\cdot F_{k}$ be a minimal
fragmentation of rank $\alpha\geq 1$ of a word $X\in\mathcal{R}_{\alpha}$.

  1. (i)

Let $F_{i}$ be a truncated fragment of rank $\alpha$ in $\mathcal{F}$. Then
$|F_{i}|_{\alpha-1}\geq\frac{1}{\zeta}$ and $F_{i}=uFv$ where $F$ is a
fragment of rank $\alpha$, $F_{i}\sim F$,
$|u|_{\alpha-1},|v|_{\alpha-1}<\zeta$ and the base $\mathsf{P}$ for the
corresponding fragment $\mathsf{F}$ in $\Gamma_{\alpha-1}$ satisfies
$|\mathsf{P}|_{\alpha-1}>13$.

  2. (ii)

If $K$ is a fragment of rank $\alpha$ in $X$ and $\mu_{\mathrm{f}}(K)\geq
3\lambda+15\omega$ then $F_{i}\sim K$ for some $i$.

  3. (iii)

Let $X=P_{0}K_{1}P_{1}\dots K_{r}P_{r}$ where $K_{i}$ are fragments of rank
$\alpha$ with $\mu_{\mathrm{f}}(K_{i})\geq 3\lambda+13\omega$ for all $i$.
Then there exists another minimal fragmentation $\mathcal{F}^{\prime}$ of rank
$\alpha$ of $X$ such that each $K_{i}$ is contained in a compatible truncated
fragment of rank $\alpha$ in $\mathcal{F}^{\prime}$.

###### Proof.

(i) If $|F_{i}|_{\alpha-1}<\frac{1}{\zeta}$ then we could replace $F_{i}$ by
its fragmentation of rank $\alpha-1$ which would decrease the weight of
$\mathcal{F}$. By Proposition 9.21α-1 in the case $\alpha\geq 2$ (in the case
$\alpha=1$ we take $u$ and $v$ empty) we have $F_{i}=uFv$ where $F$ is a
fragment of rank $\alpha$, $F_{i}\sim F$ and
$|u|_{\alpha-1},|v|_{\alpha-1}<\zeta$. If $\mathsf{F}$ is the corresponding
fragment of rank $\alpha$ in $\Gamma_{\alpha-1}$ and $\mathsf{P}$ is the base
for $\mathsf{F}$ then by Proposition 14.1α-1

$|\mathsf{P}|_{\alpha-1}>\frac{1}{1.3}\left(\frac{1}{\zeta}-2\zeta-2.2\right)>13.$

(ii) Let $K$ be a fragment of rank $\alpha$ in $X$ and
$\mu_{\mathrm{f}}(K)\geq 3\lambda+15\omega$. We assume that there is no
truncated fragment $F_{i}$ of rank $\alpha$ such that $F_{i}\sim K$.

By Proposition 8.10 and the assumption, if $H$ is a common part of $K$ and
some $F_{i}$ of rank $\alpha$ then $H$ contains no fragment $K^{\prime}$ of
rank $\alpha$ with $\mu_{\mathrm{f}}(K^{\prime})\geq\lambda+2.6\omega$. By
Lemma 10.8, if $H$ is a common part of $K$ and some $F_{i}$ of rank
$\beta<\alpha$ then $H$ contains no fragment $K^{\prime}$ of rank $\alpha$
with $\mu_{\mathrm{f}}(K^{\prime})\geq 3.2\omega$. In particular, $K$ is not
contained in any $F_{i}$ . Let

$X\eqcirc F_{1}F_{2}\dots F_{p}^{\prime}F_{p}^{\prime\prime}\dots
F_{q}^{\prime}F_{q}^{\prime\prime}\dots F_{k}\quad\text{where}\quad
F_{p}\eqcirc F_{p}^{\prime}F_{p}^{\prime\prime},\quad F_{q}\eqcirc
F_{q}^{\prime}F_{q}^{\prime\prime},\quad K\eqcirc
F_{p}^{\prime\prime}F_{p+1}\dots F_{q}^{\prime}.$

If some $F_{i}$ is contained in $K$ and has rank $\alpha$ then by the remark
above and 14.2, $K$ is covered by at most three of the $F_{j}$’s. In this
case, by Proposition 8.11 we would have

$\mu_{\mathrm{f}}(K)\leq 3(\lambda+2.6\omega)+2\zeta\omega<3\lambda+15\omega$

contrary to the hypothesis. Therefore, each $F_{i}$ that contained in $K$ has
rank $\beta<\alpha$. Now by Proposition 8.11, $F_{p}F_{p+1}\dots F_{q}$
contains a fragment $K^{\prime}$ of rank $\alpha$ with

$\mu_{\mathrm{f}}(K^{\prime})\geq\mu_{\mathrm{f}}(K)-2(\lambda+2.6\omega)-2\zeta\omega>29\omega.$

For a base $P$ of $K^{\prime}$ we have $|P|_{\alpha-1}>29$ and by Proposition
14.1α-1, $|K^{\prime}|_{\alpha-1}>20$. This implies that
$\text{weight}_{\alpha}(F_{p}\cdot F_{p+1}\cdot\ldots\cdot F_{q})>1$ and we
get a contradiction with minimality of $\mathcal{F}$ since we can replace
$F_{p}F_{p+1}\ldots F_{q}$ in $\mathcal{F}$ by a single truncated fragment of
rank $\alpha$. This finishes the proof.

(iii) By (ii), for each $i=1,2,\dots,r$ there exists a truncated fragment
$F_{t_{i}}$ of rank $\alpha$ in $\mathcal{F}$ such that $K_{i}\sim F_{t_{i}}$.
Proposition 8.13 easily implies that $F_{t_{i}}\cup K_{i}$ is a truncated
fragment of rank $\alpha$. For each $i=1,2,\dots,r$ we consequently replace
$F_{t_{i}}$ in $\mathcal{F}$ by $F_{t_{i}}\cup K_{i}$. Since we do not
increase $\text{weight}_{\alpha}(\mathcal{F})$, the resulting fragmentation
$\mathcal{F}^{\prime}$ of $X$ is also minimal. ∎

######  14.4 Lemma.

Let $\alpha\geq 1$ and $X,Y\in\mathcal{R}_{\alpha}$ be close in rank
$\alpha-1$. Then

$|Y|_{\alpha}<1.3|X|_{\alpha}+2.2\zeta.$

###### Proof.

Let $\mathcal{F}$ be a minimal fragmentation of $X$. We represent $X$ and $Y$
by close paths $\mathsf{X}$ and $\mathsf{Y}$ in $\Gamma_{\alpha-1}$. Then
$\mathcal{F}$ induces the partition of $\mathsf{X}$, denoted
$\bar{\mathcal{F}}$, into (path) truncated fragments of ranks $\leq\alpha$.

Let

$\mathsf{X}=\mathsf{P}_{0}\mathsf{H}_{1}\mathsf{P}_{1}\dots\mathsf{H}_{r}\mathsf{P}_{r}$

where $\mathsf{H}_{1},\dots,\mathsf{H}_{r}$ are all truncated fragments of
rank $\alpha$ in $\bar{\mathcal{F}}$. If $r=0$ then
$|X|_{\alpha}=\zeta|X|_{\alpha-1}$, $|Y|_{\alpha}\leq\zeta|X|_{\alpha-1}$ and
the statement simply follows from Proposition 14.1α-1. We assume $r>0$. By
Lemma 14.3(i), for each $i$ we have
$\mathsf{H}_{i}=\mathsf{u}_{i}\mathsf{H}_{i}^{\prime}\mathsf{v}_{i}$ where
$\mathsf{H}_{i}^{\prime}$ is a fragment of rank $\alpha$,
$\mathsf{H}_{i}^{\prime}\sim\mathsf{H}_{i}$,
$|\mathsf{u}|_{\alpha-1},|\mathsf{v}|_{\alpha-1}<\zeta$, and the base
$\mathsf{S}_{i}$ for $\mathsf{H}_{i}$ satisfies
$|\mathsf{S}_{i}|_{\alpha-1}>13$. Using Proposition 10.16α-1 we find fragments
$\mathsf{H}^{\prime\prime}_{i}$ and $\mathsf{G}_{i}$ of rank $\alpha$ in
$\mathsf{X}$ and $\mathsf{Y}$ respectively where
$\mathsf{H}^{\prime}_{i}=\mathsf{w}_{i}\mathsf{H}_{i}^{\prime\prime}\mathsf{z}_{i}$,
$|\mathsf{w}_{i}|_{\alpha-1},|\mathsf{z}_{i}|_{\alpha-1}<1.15$,
$\mathsf{H}_{i}\sim\mathsf{H}_{i}^{\prime\prime}\sim\mathsf{G}_{i}$ and
$\mathsf{H}_{i}^{\prime\prime}$ and $\mathsf{G}_{i}$ are close in rank
$\alpha-1$. Using Lemma 10.13(i)α-1 after each application of Proposition
10.16α-1 we can assume that $\mathsf{G}_{i}$ are disjoint, i.e.

$\mathsf{Y}=\mathsf{Q}_{0}\mathsf{G}_{1}\mathsf{Q}_{1}\dots\mathsf{G}_{r}\mathsf{Q}_{r}.$

By Proposition 14.1α-1 we have

$\displaystyle|\mathsf{Q}_{0}|_{\alpha-1}<1.3|\mathsf{P}_{0}\mathsf{u}_{1}\mathsf{w}_{1}|_{\alpha-1}+2.2,$
$\displaystyle|\mathsf{Q}_{i}|_{\alpha-1}<1.3|\mathsf{z}_{i}\mathsf{v}_{i}\mathsf{P}_{i}\mathsf{u}_{i+1}\mathsf{w}_{i+1}|_{\alpha-1}+2.2\quad(i=1,\dots,r-1),$
$\displaystyle|\mathsf{Q}_{k}|_{\alpha-1}<1.3|\mathsf{z}_{k}\mathsf{v}_{k}\mathsf{P}_{k}|_{\alpha-1}+2.2.$

We have also

$|X|_{\alpha}=r+\zeta\sum_{i=1}^{r}|\mathsf{P}_{i}|_{\alpha-1}\quad\text{and}\quad|Y|_{\alpha}\leq
r+\zeta\sum_{i=1}^{r}|\mathsf{Q}_{i}|_{\alpha-1}.$

Then

$\displaystyle|Y|_{\alpha}$
$\displaystyle<r+1.3\zeta\sum_{i=1}^{r}|\mathsf{P}_{i}|_{\alpha-1}+1.3r\zeta(2.3+2\zeta)+2.2\zeta(r+1)$
$\displaystyle=(1+1.3\zeta(4.5+2\zeta))r+1.3\zeta\sum_{i=1}^{r}|\mathsf{P}_{i}|_{\alpha-1}+2.2\zeta$
$\displaystyle<1.3|X|_{\alpha}+2.2\zeta.$

∎

###### Proof of Proposition 14.1.

Let $X,Y\in\mathcal{R}_{\alpha}$ be close in rank $\alpha$. Let $\mathcal{F}$
be a minimal fragmentation of $X$. We consider close paths $\mathsf{X}$ and
$\mathsf{Y}$ in $\Gamma_{\alpha}$ labeled $X$ and $Y$ respectively. Then
$\mathcal{F}$ induces the partitions of $\mathsf{X}$ into (path) truncated
fragments of ranks $\leq\alpha$,

$\mathsf{X}=\mathsf{F}_{1}\cdot\mathsf{F}_{2}\cdot\ldots\cdot\mathsf{F}_{k}.$

Let $\mathsf{X}^{-1}\mathsf{u}\mathsf{Y}\mathsf{v}$ be a coarse bigon. We fix
some bridge partitions of $\mathsf{u}$ and $\mathsf{v}$. Let $\Delta$ be a
filling diagram of rank $\alpha$ with boundary loop
$\tilde{\mathsf{X}}^{-1}\tilde{\mathsf{u}}\tilde{\mathsf{Y}}\tilde{\mathsf{v}}$.
Up to switching of $\mathsf{u}$ and $\mathsf{v}$ we can assume that $\Delta$
is reduced and has a tight set $\mathcal{T}$ of contiguity subdiagrams. Let
$\mathsf{D}_{1}$, $\dots$, $\mathsf{D}_{r}$ be all cells of rank $\alpha$ of
$\Delta$. In the process of forming $\mathcal{T}$ we assume that we pick first
the contiguity subdiagrams of $\mathsf{D}_{i}$ to $\tilde{\mathsf{X}}^{-1}$
choosing them with maximal possible contiguity segment occurring in
$\tilde{\mathsf{X}}^{-1}$. Let

$\mathsf{X}=\mathsf{P}_{0}\mathsf{K}_{1}\mathsf{P}_{1}\dots\mathsf{K}_{r}\mathsf{P}_{r}\quad\text{and}\quad\mathsf{Y}=\mathsf{Q}_{0}\mathsf{M}_{1}\mathsf{Q}_{1}\dots\mathsf{M}_{r}\mathsf{Q}_{r}.$

where $\mathsf{K}_{i}$ and $\mathsf{M}_{i}$ are the corresponding active
fragments of rank $\alpha$ in $\mathsf{X}$ and $\mathsf{Y}$. By the way we
produce $\mathcal{T}$ and by Proposition 9.21α-1 in the case $\alpha\geq 2$ we
have the following:

(*) For all $i$, the fragment $\mathsf{K}_{i}$ cannot be extended in
$\mathsf{P}_{i-1}\mathsf{K}_{i}\mathsf{P}_{i}$. In particular, if $\mathsf{F}$
is a truncated fragment of rank $\alpha$ contained in
$\mathsf{P}_{i-1}\mathsf{K}_{i}\mathsf{P}_{i}$ and containing $\mathsf{K}_{i}$
then $\mathsf{F}=\mathsf{w}_{1}\mathsf{K}_{i}\mathsf{w}_{2}$ where
$|\mathsf{w}_{i}|_{\alpha-1}<\zeta$ $(i=1,2)$

By Lemma 14.3(iii) we can assume that each $\mathsf{K}_{i}$ is contained in a
compatible truncated fragment $\mathsf{F}_{t_{i}}$ of rank $\alpha$. Let

$\mathsf{X}=\mathsf{P}_{0}^{\prime}\mathsf{F}_{t_{1}}\mathsf{P}_{1}^{\prime}\dots\mathsf{F}_{t_{r}}\mathsf{P}_{r}^{\prime}.$

Note that

$|X|_{\alpha}=r+\sum_{i}|\mathsf{P}_{i}^{\prime}|_{\alpha}\quad\text{and}\quad|Y|_{\alpha}\leq
r+\sum_{i}|\mathsf{Q}_{i}|_{\alpha}.$

By (*),

$|\mathsf{P}_{i}^{\prime}|_{\alpha}\geq|\mathsf{P}_{i}|_{\alpha}-\zeta^{2}\text{
for
}i=0,r,\quad|\mathsf{P}_{i}^{\prime}|_{\alpha}\geq|\mathsf{P}_{i}|_{\alpha}-2\zeta^{2}\text{
for }1\leq i\leq r-1.$

Hence

(14-1) $|X|_{\alpha}\geq r+\sum_{i}|\mathsf{P}_{i}|_{\alpha}-2r\zeta^{2}.$

We give an upper bound on $|\mathsf{Q}_{i}|_{\alpha}$ in terms of
$|\mathsf{P}_{i}|_{\alpha}$. First we consider the case $1\leq i\leq r-1$.
There are three possibilities for the subdiagram of $\Delta$ surrounded by
$\mathsf{D}_{i}$ and $\mathsf{D}_{i+1}$ and contiguity subdiagrams of
$\mathsf{D}_{i}$ and $\mathsf{D}_{i+1}$ to $\tilde{\mathsf{X}}^{-1}$ and
$\tilde{\mathsf{Y}}$, depending on the presence of contiguity subdiagrams from
$\mathcal{T}$ (see Figure 38).

$\mathsf{D}_{i}$
---
$\mathsf{D}_{i+1}$
---
$\tilde{\mathsf{X}}$
---
$\tilde{\mathsf{Y}}$
---
(a)
---
(b)
---
(c)
---
Figure 38.

Note that according to Definition 6.12, all the components of
$\Delta-\cup_{\Pi\in\mathcal{T}}$ are small diagrams of rank $\alpha-1$, so we
can use bounds from Proposition 7.12α-1. In cases (a) and (b) we have
$|\mathsf{Q}_{i}|_{\alpha}\leq 6\zeta^{2}\eta<0.6\zeta$ and
$|\mathsf{Q}_{i}|_{\alpha}\leq 4\zeta^{2}\eta<0.4\zeta$ respectively. Assume
that case (c) holds. Then
$\mathsf{P}_{i}=\mathsf{u}_{1}\mathsf{S}\mathsf{u}_{2}$ and
$\mathsf{Q}_{i}=\mathsf{v}_{1}\mathsf{T}\mathsf{v}_{2}$ where $\mathsf{S}$ and
$\mathsf{T}$ are close in rank $\alpha-1$ and
$|\mathsf{u}_{i}|_{\alpha},|\mathsf{v}_{i}|_{\alpha}\leq
4\zeta^{2}\eta<0.4\zeta$. Using Lemma 14.4, we get

$|\mathsf{Q}_{i}|_{\alpha}<1.3|\mathsf{P}_{i}|_{\alpha}+3\zeta$

Note that this inequality holds also in cases (a) and (b).

Now let $i=0$ or $i=r$. If $r>0$ then the difference of the case $i=0$ from
the case $1\leq i\leq r-1$ is that we can have an extra contiguity subdiagram
between $\mathsf{Y}$ and the central arc of $\tilde{\mathsf{u}}$ (see Figure
39).

$\tilde{\mathsf{X}}$
---
$\tilde{\mathsf{Y}}$
---
$\tilde{\mathsf{u}}$
---
$\mathsf{D}_{1}$
---
Figure 39.

We then have

$|\mathsf{Q}_{0}|_{\alpha}<1+1.3|\mathsf{P}_{0}|_{\alpha}+3\zeta$

and, similarly,

$|\mathsf{Q}_{r}|_{\alpha}<1+1.3|\mathsf{P}_{r}|_{\alpha}+3\zeta.$

If $r=0$ we have a single bound instead,

$|\mathsf{Q}_{0}|_{\alpha}<2+1.3|\mathsf{P}_{0}|_{\alpha}+3\zeta.$

Summarizing, with (14-1) we get

$\displaystyle|Y|_{\alpha}$ $\displaystyle\leq
r+\gamma\sum_{i}|\mathsf{P}_{i}|_{\alpha}+2+3\zeta(r+1)$
$\displaystyle=(1+3\zeta)r+1.3\sum_{i}|\mathsf{P}_{i}|_{\alpha}+2+3\zeta$
$\displaystyle<1.3|X|_{\alpha}+2.2.$

∎

######  14.5 Corollary.

If $F$ is a fragment of rank $\alpha$ and $\mu_{\mathrm{f}}(F)\geq t\omega$
then $|F|_{\alpha-1}>\frac{1}{1.3}(t-2.2)$. In particular,
$|F|>\frac{1}{1.3}{\zeta^{1-\alpha}}(t-2.2)$.

######  14.6 Corollary.

Let $Y=u_{1}X_{1}u_{2}X_{2}u_{3}$ in $\Gamma_{\alpha}$ where
$X_{i},Y\in\mathcal{R}_{\alpha}$ and $u_{i}\in\mathcal{H}_{\alpha}$. Then
$|Y|_{\alpha}\leq 1.3(|X_{1}|_{\alpha}+|X_{2}|_{\alpha})+4.8$.

###### Proof.

Follows from Propositions 9.19(i) and 14.1. ∎

The following two statements are proved under the assumption that a normalized
presentation (2-1) of $G$ satisfies the iterated small cancellation condition
(S0)–(S3) for all $\alpha\geq 1$. We therefore will be assuming that all
statements starting from Section 5 hold for all values of $\alpha$.

######  14.7 Proposition.

Let $W$ be a word with $|W|\leq\alpha$ and let $W=X$ in $G_{\alpha}$ where
$X\in\mathcal{R}_{\alpha}$. Then $|X|_{\alpha}<0.3$, $X$ contains no fragments
$F$ of rank $\beta>\alpha$ with $\mu_{\mathrm{f}}(F)\geq 3\omega$ and, in
particular, $X\in\cap_{\alpha\geq 1}\mathcal{R}_{\alpha}$.

By Corollary 14.5 it is enough to prove that $|X|_{\alpha}<0.3$. We proceed by
induction on $\alpha$. If $\alpha=1$ then $X$ is the freely reduced form of
$W$ and $|X|_{1}\leq\zeta|X|<0.3$. Let $\alpha>1$. Let $W\eqcirc W_{1}a$,
$a\in\mathcal{A}^{\pm 1}$ and $W_{1}=X_{1}$ in $G_{\alpha-1}$ where
$X_{1}\in\mathcal{R}_{\alpha-1}$. By Corollary 14.5, the inductive hypothesis
and Proposition 9.15, equality $X=X_{1}a$ holds already in $G_{\alpha-1}$. By
Corollary 14.6α-1

$|X|_{\alpha}\leq\zeta|X|_{\alpha-1}\leq\zeta(1.3(0.3+0.3)+4.8)<0.3.$

######  14.8 Corollary.

Every element of $G$ can be represented by a word $X$ reduced in $G$ such that
for some $\alpha\geq 1$, $X$ contains no fragments $F$ of rank
$\beta\geq\alpha$ with $\mu_{\mathrm{f}}(F)\geq 3\omega$.

##  15\. A graded presentation for the Burnside group

In this section we show that for sufficiently large odd $n$ the Burnside group
$B(m,n)$ has a graded presentation which satisfies the iterated small
cancellation condition formulated in Section 2.

We fix an odd number $n>2000$. We are going to construct a graded presentation
of the form

(15-1) $\bigl{\langle}\mathcal{A}\;\bigm{|}\;C^{n}=1\ (C\in\bigcup_{\alpha\geq
1}\mathcal{E}_{\alpha})\bigr{\rangle}$

where all relators of all ranks $\alpha$ are $n$-th powers. We assume that
values of the parameters $\lambda$ and $\Omega$ are chosen as in Theorem 3,
i.e.

$\lambda=\frac{80}{n},\quad\Omega=0.25n.$

We will use also the following extra parameters:

$p_{0}=39,\quad p_{1}=p_{0}+26=65.$

In what follows, we define the set $\mathcal{E}_{\alpha+1}$ under the
assumption that sets $\mathcal{E}_{\beta}$ are already defined for all
$\beta\leq\alpha$. We fix the value of rank $\alpha\geq 0$ and assume that the
presentation (15-1) satisfies small cancellation conditions (S0)–(S3) in 2.8,
2.9 and in normalized in the sense Definition 2.10 for all values of the rank
up to $\alpha$.

We can therefore assume that all statements in Sections 5–13 are true for the
current value of $\alpha$ and below.

According to Propositions 11.5 and 11.13 each element of infinite order of
$G_{\alpha}$ is conjugate to a power of a simple period over $G_{\alpha}$. We
will define $\mathcal{E}_{\alpha+1}$ as a certain set of simple periods over
$G_{\alpha}$. This will automatically imply condition (S0) with
$\alpha:=\alpha+1$.

Since $n$ is odd, by Corollary 13.11 we obtain also that (S3) holds with
$\alpha:=\alpha+1$.

Before going to the chain of definitions in the next section, we formulate the
following two conditions (P1) and (P2) on $\mathcal{E}_{\alpha+1}$ (which can
be viewed as ‘‘periodic’’ versions of (S1) and (S2) for the value of rank
$\alpha:=\alpha+1$).

  * (P1)

For each $A\in\mathcal{E}_{\alpha+1}$, $[A]_{\alpha}\geq 0.25$.

  * (P2)

Let $\mathsf{L}_{1}$ and $\mathsf{L}_{2}$ be periodic lines in
$\Gamma_{\alpha}$ with periods $A,B\in\mathcal{E}_{\alpha+1}$ respectively.
Assume that a subpath $\mathsf{P}$ of $\mathsf{L}_{1}$ and a subpath of
$\mathsf{Q}$ of $\mathsf{L}_{2}$ are close and $|\mathsf{P}|\geq p_{1}|A|$.
Then $\mathsf{L}_{1}$ and $\mathsf{L}_{2}$ are parallel.

The main goal of the construction of $\mathcal{E}_{\alpha+1}$ will be to
satisfy (P1) and (P2). Note that (P1) immediately implies (S1) for
$\alpha:=\alpha+1$ because of the assumption $n>2000$. Later we prove that
(P2) implies (S2)α+1. (The difference between (P2) and (S2)α+1 is that in (P2)
we measure periodic words by the number of periods while in (S2)α+1 we use the
length function $|\cdot|_{\alpha}$. An appropriate bound will be given in
Proposition 16.6.)

Our first step is to define a set of simple periods over $G_{\alpha}$ which
potentially violate (P2) (they will be excluded in the definition of
$\mathcal{E}_{\alpha+1}$).

######  15.1 Definition.

A simple period $A$ over $G_{\alpha}$ is suspended of level 0 if there exist a
simple period $B$ not conjugate in $G_{\alpha}$ to $A$ and words
$P\in\operatorname{Per}(A)$ and $Q\in\operatorname{Per}(B)$ such that $P$ and
$Q$ are close in $G_{\alpha}$ and $|Q|\geq p_{1}|B|$.

At first sight, we could simply define $\mathcal{E}_{\alpha+1}$ by excluding
periods $A$ as in Definition 15.1 from the set of all simple periods over
$G_{\alpha}$. However, in this case we cannot guarantee a necessary lower
bound on $[A]_{\alpha}$ for $A\in\mathcal{E}_{\alpha+1}$ in (P1). Roughly
speaking, we need to claim that a fragment of rank $\beta\leq\alpha$ can cover
only a ‘‘small’’ part of a periodic word with a period
$A\in\mathcal{E}_{\alpha+1}$; moreover, we need an exponentially decreasing
upper bound on the size of this part when $\beta$ decreases (compare with the
definition of the function $|\cdot|_{\alpha}$ in 2.7). To achieve this, we
enlarge the set of excluded simple periods over $G_{\alpha+1}$ by adding
potentially ‘‘bad’’ examples of this sort.

######  15.2 Definition.

A simple period $A$ over $G_{\alpha}$ is suspended of level $m\geq 1$ if there
exist a suspended period $B$ of level $m-1$ not conjugate to $A$ in
$G_{\alpha}$, and a reduced in $G_{\alpha}$ word of the form $XQY$ such that
$Q\in\operatorname{Per}(B)$, $|Q|\geq 4|B|$ and $XQY$ is close in $G_{\alpha}$
to a word $P\in\operatorname{Per}(A)$.

######  15.3 Definition.

Let $\mathcal{P}_{\alpha}$ denote the set of all simple periods over
$G_{\alpha}$ and $\mathcal{S}_{\alpha}$ denote the set of all suspended simple
periods over $G_{\alpha}$ of all levels $m\geq 0$. For
$\mathcal{E}_{\alpha+1}$ we take any set of representatives of equivalence
classes in $\mathcal{P}_{\alpha}\setminus\mathcal{S}_{\alpha}$ with respect to
the equivalence

$A\sim B\Leftrightarrow A\text{ is conjugate to $B^{\pm 1}$ in }G_{\alpha}.$

The definition implies that any simple period over $G_{\alpha}$ in
$\mathcal{P}_{\alpha}\setminus\mathcal{S}_{\alpha}$ has finite order in
$G_{\alpha+1}$. Since $\mathcal{P}_{\alpha+1}\subseteq\mathcal{P}_{\alpha}$,
it follows that any simple period over $G_{\alpha+1}$ and, in particular, any
word in $\mathcal{E}_{\beta}$ for $\beta\geq\alpha+1$ belongs to
$\mathcal{S}_{\alpha}$. As a consequence, we prove now that a fragment of rank
$\alpha+1$ cannot cover a large periodic word with a simple period $A$ over
$G_{\alpha+1}$. (So here is the trick: the definition of the set of suspended
periods over $G_{\alpha}$ of levels $m\geq 1$ serves condition (P1) for the
future rank $\alpha+1$.)

######  15.4 Remark.

By construction, we obtain a normalized presentation (15-1) (see Definition
2.10).

######  15.5 Proposition.

Let $A$ be a simple period over $G_{\alpha+1}$. If an $A$-periodic word $P$ is
a subword of a fragment of rank $\alpha+1$ then $|P|<4|A|$.

###### Proof.

As observed above, $A\in\mathcal{S}_{\alpha}$. Let $UPV$ be a fragment of rank
$\alpha+1$ where $P\in\operatorname{Per}(A)$. Then $UPV$ is close in
$G_{\alpha}$ to a word $Q\in\operatorname{Per}(B)$ where
$B\in\mathcal{E}_{\alpha+1}$. Since $A$ is of infinite order in
$G_{\alpha+1}$, it is not conjugate to $B$ in $G_{\alpha}$. In this case,
Definition 15.2 says that if $|P|\geq 4|A|$ then $B\in\mathcal{S}_{\alpha}$
which would contradict Definition 15.3. ∎

Proposition 15.5 with $\alpha:=\alpha-1$ is an important but not sufficient
ingredient in the proof of (P1). We need also to ensure that if a subword of
fragment of rank $\beta<\alpha$ is a subword of an $A$-periodic word with
$A\in\mathcal{E}_{\alpha+1}$ then its length compared to $|A|$ is
‘‘exponentially decreasing when $\beta$ decreases’’. We prove a precise form
of this statement in the next section by showing that coarsely periodic words
have a certain property of hierarchical containment: a coarsely $A$-periodic
word $S$ over $G_{\alpha}$ has $t$ disjoint occurrences of coarsely periodic
words over $G_{\alpha-1}$ with sufficiently large number of periods where $t$
is approximately the number of periods $A$ in $S$.

##  16\. Hierarchical containment of coarsely periodic words

Starting from this point, all statements are formulated and proved under
assumption that the group $G$ has a specific presentation (15-1) defined in
Section 15. The goal of this section is to prove the following property of
suspended periods over $G_{\alpha}$ and to finalize the proof of the fact that
the presentation (15-1) satisfies conditions (S0)–(S3). As in Section 15 we
assume fixed the value of rank $\alpha\geq 0$ and assume that the normalized
presentation (15-1) satisfies conditions (S0)–(S3) for ranks less or equal
$\alpha$; so we can use all statements in Sections 5–15 for any rank up to
$\alpha$.

######  16.1 Proposition.

Let $A$ be a suspended period over $G_{\alpha}$. Then there exists a simple
period $B$ over $G_{\alpha}$ such that:

  1. (i)

A cyclic shift of $A$ contains a coarsely $B$-periodic word $T$ over
$G_{\alpha}$ with $\ell_{B}(T)\geq p_{0}$.

  2. (ii)

Moreover, this subword $T$ has the following property. Let $\mathsf{S}$ be a
coarsely $A$-periodic segment in $\Gamma_{\alpha}$ with
$\ell_{A}(\mathsf{S})\geq 4$. Then there are an $A$-periodic base $\mathsf{P}$
for $\mathsf{S}$, $\ell_{A}(\mathsf{S})-3$ translates $\mathsf{T}$,
$s_{A,\mathsf{P}}\mathsf{T}$, $\dots$,
$s_{A,\mathsf{P}}^{\ell(\mathsf{S})-4}\mathsf{T}$ of a coarsely $B$-periodic
segment $\mathsf{T}$ in $\mathsf{P}$ with
$\operatorname{\mathit{label}}(\mathsf{T})\eqcirc T$ and
$\ell_{A}(\mathsf{S})-3$ disjoint coarsely $B$-periodic segments
$\mathsf{V}_{i}$ $(i=0,1,\dots,\ell(\mathsf{S})-4)$ in $\mathsf{S}$ such that
$\mathsf{V}_{i}\approx s_{A,\mathsf{P}}^{i}\mathsf{T}$ for all $i$.

We start with showing how Proposition 16.1α-1 implies (P1) in the case
$\alpha\geq 1$.

######  16.2 Lemma.

Let $A$ be a simple period over $G_{\alpha}$ and let $\mathsf{S}$ and
$\mathsf{V}_{i}$ $(i=0,1,\dots,\ell_{A}(\mathsf{S})-4)$ be as in Proposition
16.1α-1. Then for any $i$, $\mathsf{V}_{i}\cup\mathsf{V}_{i+4}$ is not
contained in a fragment of rank $\alpha$.

###### Proof.

As in Proposition 16.1α-1, let $\mathsf{P}$ be an $A$-periodic base for
$\mathsf{S}$ in $\Gamma_{\alpha-1}$ containing $t-3$ translates $\mathsf{T}$,
$s_{A,\mathsf{P}}\mathsf{T}$, $\dots$, $s_{A,\mathsf{P}}^{t-4}\mathsf{T}$
where $\mathsf{T}$ is a coarsely periodic segment with another period $B$ and
$\ell_{B}(\mathsf{T})\geq p_{0}$. Assume that a fragment $\mathsf{K}$ of rank
$\alpha$ in $\Gamma_{\alpha-1}$ contains $\mathsf{V}_{i}$ and
$\mathsf{V}_{i+4}$. Let $\mathsf{L}$ be the base axis for $\mathsf{K}$, so
$\mathsf{L}$ is a $C$-periodic line with $C\in\mathcal{E}_{\alpha}$. Denoting
$\mathsf{V}_{i}^{*}$ the stable part of $\mathsf{V}_{i}$, by Proposition
12.14α-1 we find $\mathsf{W}$ and $\mathsf{W}^{\prime}$ in $\mathsf{L}$ such
that $\mathsf{W}\approx\mathsf{V}_{i}^{*}$ and
$\mathsf{W}^{\prime}\approx\mathsf{V}_{i+4}^{*}$. Then
$\mathsf{W}\cup\mathsf{W}^{\prime}$ is close to
$s_{A,\mathsf{P}}^{i}\mathsf{T}^{*}\cup s_{A,\mathsf{P}}^{i+4}\mathsf{T}^{*}$.
Since $A\in\mathcal{S}_{\alpha-1}$, according to Definition 15.2α-1 this
should imply $C\in\mathcal{S}_{\alpha-1}$, a contradiction. ∎

######  16.3 Lemma.

Let $\alpha\geq 1$. Assume that a (linear or cyclic) word $X$ has $r$ disjoint
occurrences of coarsely $A$-periodic words $U_{i}$ $(i=1,\dots,r)$ over
$G_{\alpha-1}$ with $\ell_{A}(U_{i})\geq p_{0}$. Then $|X|_{\alpha-1}\geq 5r$.

###### Proof.

The statement is immediate if $\alpha=1$. Assume that $\alpha>1$.

Consider a fragmentation $\mathcal{F}$ of rank $\alpha-1$ of $X$ (definition
2.7). Let $S_{1}$, $\dots$, $S_{k}$ be the subwords of fragments of rank
$\alpha-1$ in $\mathcal{F}$. By Proposition 16.1α-1 each $U_{i}$ contains
$p_{0}-3=36$ disjoint coarsely $B$-periodic words $V_{i,j}$ $(j=1,\dots,36)$
over $G_{\alpha-2}$ with $\ell_{B}(V_{i,j})\geq p_{0}$. We can assume that
$U_{i}$ and $V_{i,j}$ are indexed in their natural order from the start to the
end in $X$. By Lemma 16.2, each $S_{i}$ intersects at most 6 consequent
subwords $V_{i,j},V_{i,j+1},\dots,V_{i,j+5}$. Excluding $V_{i,j}$ with $1\leq
j\leq 6$, we obtain that each $S_{i}$ intersects at most 6 of all the
remaining $V_{i,j}$. By induction, we conclude that

$|X|_{\alpha-1}\geq k+5\zeta\max\\{0,\>30r-6k\\}$

With fixed $r$, the minimal value of the right-hand side is achieved when
$30r-6k=0$. This gives the bound $|X|_{\alpha-1}\geq 5r$. ∎