\section{Improving nonexponential strata}\label{improvesec}

In \cite{tits1}, the authors improve the behavior of nonexponential
strata in a nonconstructive fashion.  We retrace some of their steps
here, replacing the nonconstructive parts by constructive arguments.

Let $H_r=\{E_r\}$ be a nonexponential stratum of a normalized train
track map $f\co G\rightarrow G$, and let $\rho$ be a path in $G_{r-1}$
originating at the terminal vertex of $E_r$.  We define a new map $f'\co
G'\rightarrow G'$ by removing $E_r$ and adding an edge $E_r'$ whose
initial vertex is the initial endpoint of $E_r$ and whose terminal
vertex is the terminal vertex of $\rho$.  We obtain $u_r'$ by tightening
$\bar{\rho}u_r(\rho f)$, so that $E_r'f'=E_r'u_r'$.  There is an obvious
homotopy equivalence $g\co G\rightarrow G'$ that sends $E_r$ to
$E_r'\bar{\rho}$.  With this marking, $f'$ induces the same outer
automorphism as $f$.  We say the $E_r'$ is obtained from $E_r$ by {\em
sliding along $\rho$}.

Let $\tilde{f}\co \tilde{G}\rightarrow \tilde{G}$ be a lift of $f$ that
fixes the initial endpoint of a lift $\tilde{E_r}$ of $E_r$.
Then $\tilde{f}$ leaves invariant a copy $H$ of the universal cover of
the connected component of $G_{r-1}$ that contains $u_r$.  Let
$h=\tilde{f}|_H$, and let $v_0$ be the terminal endpoint of
$\tilde{E}_r$.  Note that $v_0\in H$, and that $[v_0, v_0h]$ projects to
$u_r$.

\begin{lem}\label{constlem}
There exists a slide of $E_r$ to $E_r'$ with $E_r'f'=E_r'$ if and only
if $h$ fixes a point in $H$.
\end{lem}

\begin{proof}
If $h$ fixes $v\in H$, then sliding $E_r$ along $[v_0, v]$ yields a
fixed edge $E_r'$.  Conversely, if there exists a path $\rho$ such that
sliding $E_r$ along $\rho$ yields a fixed edge, then the terminal
endpoint of the lift of $\rho$ is fixed by $h$.
\end{proof}

In \secref{fixedsec}, we present an algorithm for detecting fixed points
of $h$.

\begin{lem}\label{nofixlem}
Assume that $h$ has no fixed points.  Let $U_{k}=[v_0, v_0h^k]$ and
$V_k=U_{k}\wedge U_{k+1}$ for $k\geq 0$.  Then $V_k$ is a proper
prefix of $V_{k+1}$.
\end{lem}

\begin{proof}
This follows from the discussion of preferred edges in the proof of
\cite[Proposition~5.4.3]{tits1}.
\end{proof}

As an immediate consequence of \lemref{nofixlem}, we obtain the
following lemma.
\begin{lemnp}\label{periodicpt}
If $h$ has a periodic point, then $h$ has a fixed point.
\end{lemnp}

The following proposition is the main result of this section; it
replaces a nonconstructive argument in \cite{tits1}.

\begin{prop}\label{makeeff}
Assume that $h$ has no fixed points.  We can compute a vertex in $v\in
H$ and an exponent $m\geq 1$ such that sliding $E_r$ along $[v_0, v]$
yields $E_r'(f^m)'=E_r'\cdot u_r'$ and $u_r'$ is a closed path starting
and ending at a fixed vertex.
\end{prop}

\begin{proof}
Let $v_k$ equal the terminal vertex of the path $V_k$
(\lemref{nofixlem}),\footnote{This agrees with our original definition
of $v_0$.} and let $w_k=[v_k, v_{k+1}]$.  The path $w_{k+m}$ is a
subpath of $w_kh^m$ for all $k, m\geq 0$.

The idea of the proof is to compute $w_0, w_1, w_2,\ldots, w_k$ until we
identify a suitable vertex $v$ in $w_k$.  Since $w_{k+1}$ is a subpath
of $w_kh$, we have $height(w_{k+1})\leq height(w_k)$, so that the height
of the paths $w_k$ has to stabilize eventually.  The following procedure
assumes that the height remains constant;  should the height drop while
the procedure is in progress, we simply start over.

Assume the height stabilizes at $r$.  This means that $H_r$ cannot be a
zero stratum.  Now, if $H_r$ is nonexponential, we have $|w_{k+1}|_r\leq
|w_k|_r$.  We keep iterating until we find $w_k$ such that
$|w_k|_r=|w_{k+1}|_r\geq 1$.  Let $v$ be the initial endpoint of an
occurrence of $E_r$ in $w_k$.  Then $v$ has the desired properties (and
we do not need to replace $f$ by a higher power in this case).

Now, assume that $H_r$ is exponential.  If we encounter a path $w_k$
that contains an $r$-legal subpath of $r$-length at least $2(C_f+1)$,
then $w_{k+1}$ contains a vertex $v$ that projects to a fixed vertex of
$f$ and whose $r$-distance from the closest $r$-illegal turn is at least 
$C_f$.  Now \lemref{bcc} yields that $v$ has the desired properties.

Assume that the length of $r$-legal subpaths remains bounded below
$2(C_f+1)$.  The number of illegal turns cannot go up and must stabilize
eventually, so that eventually we will end up in the third case of
\lemref{tricho} and see a composition of Nielsen paths of height $r$ and
paths in $G_{r-1}$.  We can detect this case in a brute-force fashion,
by checking all subpaths of $w_k$ in order to see whether they are
Nielsen.

Let $v$ be the initial point of one of the Nielsen paths.  Then $v$ is
periodic of period $m$, so that sliding $E_r$ along $[v_0, v]$ yields
the desired improvement of $f^m$.
\end{proof}

\begin{defn}\label{effdef}
Let $f\co G\rightarrow G$ be a normalized relative train track map with
a nonexponential stratum $H_r=\{E_r\}$.  We say that $H_r$ is {\em
efficient} if
\begin{enumerate}
\item $E_rf$ splits as $E_r\cdot u_r$ and $u_r$ is a closed path in
$G_{r-1}$,
\item if $u_r$ is a periodic Nielsen path, then its period is one (in
this case, we say that $E_r$ is {\em linear}), and
\item if $u_r$ is nontrivial, then there exists no slide of $E_r$ to
$E_r'$ such that $E_r'f'=E_r'$.
\end{enumerate}
We say that a relative train track map is {\em efficient} if it is
normalized, all its nonexponential strata are efficient, and the
nonexponential strata are sorted in such a way that if $u_r$ and $u_s$
are of the same height but $u_r$ is Nielsen and $u_s$ is not, then
$s>r$.
\end{defn}

\begin{lem}\label{linlem}
There exists a slide of $E_r$ to $E_r'$ with $E_r'f'=E_r'u_r'$ and
$u_r'$ a periodic Nielsen path if and only if $h$ commutes with a
nontrivial deck transformation.
\end{lem}

\begin{proof}
This lemma follows from \cite[Proposition~5.4.3]{tits1}.
\end{proof}

\begin{rem}
\lemref{linlem} implies that if $H_r$ is efficient and $u_r$ is
nontrivial and non-Nielsen, then there exists no slide that takes $u_r$
to a periodic Nielsen path.
\end{rem}

An infinite ray $\rho$ starting at a fixed vertex $v_0$ is a {\em fixed
ray} if $\rho f=\rho$.  It is {\em attracting} if there exists some $N$
such that if $\eta$ is a ray starting at $v_0$ and $|\rho\wedge\eta|>N$,
then $\eta f^n$ converges to $\rho$, i.e., $|\rho\wedge \eta f^n|$ goes
to infinity.  A {\em repelling} fixed ray is an attracting fixed ray for
a homotopy inverse of $f$.  See \cite{bdrydyn} for a detailed discussion
attracting and repelling fixed points for free group automorphisms.

\begin{lem}\label{attrlem}
Let $f\co G\rightarrow G$ be an efficient relative train
track map with a nonexponential stratum $H_r=\{E_r\}$ that is neither
linear nor constant.  Let
\[
R_r=E_ru_r(u_rf)(u_rf^2)\ldots
\]
Then $R_r$ is the unique attracting fixed ray of the form $E_r\gamma$,
for $\gamma\subset G_r$, and there are no Nielsen paths of the form
$E_r\gamma$.  In particular, we have $\lim_{k\rightarrow\infty}
\rho f^k=R_r$ for all basic paths $\rho$ of height $r$.
\end{lem}

\begin{proof}
This lemma follows from the proof of \cite[Lemma~5.5.1]{tits1}.  The
assumptions of \cite{tits1} are stronger that our assumptions, but a
close inspection of the proof shows that only our assumptions are needed
for the results that we use here.
\end{proof}

If $\rho$ is a path starting and ending at fixed points, then we can
find at most one path $\rho'$ with the same endpoints such that
$\rho'f=\rho$.  In this case, we write $\rho'=\rho f^{-1}$.  We define
$\rho f^{-k}$ in the obvious fashion.  If $\rho$ is closed, then $\rho
f^{-k}$ exists for all $k$.

\begin{lem}\label{replem}
Let $f\co G\rightarrow G$ be an efficient relative train
track map with a nonexponential stratum $H_r=\{E_r\}$ that is neither
linear nor constant.  Let
\[
S_r=E_r(\bar{u}_r f^{-1})(\bar{u}_r f^{-2})\ldots
\]
Then $S_r$ is the unique repelling fixed ray of the form $E_r\gamma$,
for $\gamma\subset G_r$.  In particular, we have
$\lim_{k\rightarrow\infty} \rho f^{-k}=S_r$ for all basic paths $\rho$ of
height $r$.
\end{lem}

\begin{proof}
\lemref{attrlem} implies that $h$ only has one repelling fixed ray.
Since $S_r$ is clearly fixed, it is the unique repelling fixed ray.
\end{proof}
{"config": "arxiv", "file": "0806.2889/improve.tex"}