))=aW^\eta_La^{-1}$. Moreover, $b$ is of minimal length in $bW_{K_i}$, as $bC_0=a^{-1} D_{i-1}=\proj_{bR_{K_i}}(C_0)$ (because $a^{-1} D_{i-1}^\eta=C_0^\eta$). Hence Proposition~\ref{prop:phisurj_and_phiinvedgeedge}(2) implies that the chamber $D_i\in R_x\cap\CMin(w)$ opposite $D_{i-1}$ in $R_{K_i}(D_{i-1})$ satisfies $I_w(D_i)=\op_{L_i}(I_w(D_{i-1}))=\op_{L_i}(I_{i-1})=I_i$, as desired. 

We conclude that $D_k\in R_x\cap\CMin(w)$ is such that $I_w(D_k)=I_k=I_w(D)$. In particular, $\Cyc(\pi_w(D_k))=\Cyc(\pi_w(D))$ by Proposition~\ref{prop:R2R3distinguishccc}(3)$\Rightarrow$(1). As $D_0,D_k\in R_x$ and $\Cyc(\pi_w(C))=\Cyc(\pi_w(D_0))$, the statement (2) now follows from Lemma~\ref{lemma:ETC_geometric}(2).
\end{proof}






\section{The \texorpdfstring{$P$-splitting}{P-splitting} of an element}\label{section:TPSOAE}
Throughout this section, we fix a Coxeter system $(W,S)$ and an element $w\in \Aut(\Sigma)$ of infinite order.

In this short section, we introduce certain decompositions of $w$, which may be thought of as analogues of the decomposition of an affine isometry as a product of a rotation and of a translation. These decompositions will be used (and further investigated) in two important special cases in \S\ref{section:ICG} and \S\ref{section:ACG}.

\begin{definition}\label{definition:wparabolicsubgroup}
Call a parabolic subgroup $P$ of $W$ a {\bf $w$-parabolic subgroup}\index{parabolicsubgroup@$w$-parabolic subgroup} if there exists a $w$-essential point $x\in\Min(w)$ such that $P=\Fix_W(x)=\Stab_W(R_x)$, or equivalently, if $P$ is the stabiliser of a $w$-residue.
\end{definition}

\begin{example}\label{example:PsplittingPwmin}
By Lemma~\ref{lemma:generic_axis}, the parabolic subgroup\index[s]{Pwmin@$P_w^{\min}$ (pointwise fixer of $\Min(w)$)} $$P_w^{\min}:=\Fix_W(\Min(w))$$ is a $w$-parabolic subgroup: it is in fact the smallest $w$-parabolic subgroup, as every $w$-parabolic subgroup is the fixer of a $w$-axis.
\end{example}

Here is another source of examples of $w$-parabolic subgroups.
\begin{lemma}\label{lemma:Pmaxwparabolic}
Let $P$ be a maximal spherical parabolic subgroup normalised by $w$. Then $P$ is a $w$-parabolic subgroup. 
\end{lemma}
\begin{proof}
Let $M$ be the set of walls of $P$, and let $Z:=\bigcap_{m\in M}m\subseteq X$. By assumption, $Z$ is a nonempty closed convex subset of $X$ stabilised by $w$, and hence contains a $w$-axis $L$ (as $L$ is the convex closure of the $\langle w\rangle$-orbit of any of its points). In particular, the $w$-parabolic subgroup $P':=\Fix_W(L)$ contains $P=\Fix_W(Z)$. By maximality of $P$, we then have $P=P'$.
\end{proof}

\begin{lemma}\label{lemma:waxisawaxis}
Let $P$ be a $w$-parabolic subgroup, and $x\in\Min(w)$ with $P=\Fix_W(x)$. Then for any $a\in P$, the $w$-axis through $x$ is also an $aw$-axis.
\end{lemma}
\begin{proof}
Let $L_x$ be the $w$-axis through $x$. Then for any $a\in P$, we have $(aw)^{-1} x=w^{-1} x\in L_x$ and $awx=w\cdot w^{-1} aw x=wx\in L_x$ (as $w^{-1} aw\in P$), and hence $L_x$ is also an $aw$-axis (see e.g. \cite[Chapter II, Proposition~1.4(2)]{BHCAT0}).
\end{proof}

\begin{definition}\label{definition:Preduced}
Let $P$ be a spherical parabolic subgroup of $W$. Call an element $v\in \Aut(\Sigma)$ {\bf $P$-reduced}\index{reduced@$P$-reduced} if $C_0$ and $vC_0$ lie on the same side of every wall of $P$. 

For instance, if $P=W_I$ for some $I\subseteq S$, then $v$ is $P$-reduced if and only if $v$ is of minimal length in $W_Iv$.
\end{definition}
 

\begin{lemma}\label{lemma:PvinvPvreduced}
Let $R$ be a spherical residue such that $w$ normalises $P:=\Stab_W(R)$, and let $v\in W$ be such that $vC_0=\proj_R(C_0)$. Then $w$ is $P$-reduced if and only if $v^{-1} wv$ is $v^{-1} Pv$-reduced.
\end{lemma}
\begin{proof}
By assumption, $C_0$ and $vC_0$ lie on the same side of every wall of $P$. Since $w$ stabilises this set of walls, $wC_0$ and $wvC_0$ also lie on the same side of every wall of $P$. Hence $w$ is $P$-reduced if and only if $vC_0$ and $wvC_0$ lie on the same side of every wall of $P$, that is, if and only if $v^{-1} wv$ is $v^{-1} Pv$-reduced.
\end{proof}


Recall from \cite{straight} that an element $v\in \Aut(\Sigma)$ is called\index{Straight} {\bf straight} if $\ell(v^n)=n\ell(v)$ for all $n\in\mathbb{N}$. We now associate to each $w$-parabolic subgroup $P$ a decomposition of $w$ as a product of an element $w_0\in P$ with a $P$-reduced element $w_{\infty}\in W$, which intuitively can be thought of as the ``torsion part'' and ``straight part'' of $w$ with respect to $P$, respectively. 

\begin{prop}\label{prop:Psplitting}
Let $P$ be a $w$-parabolic subgroup. Then there are uniquely determined elements\index[s]{w0P@$w_0(P)$ (torsion part of $w$ wrt $P$)}\index[s]{winfP@$w_0(P)$ (straight part of $w$ wrt $P$)} $w_0=w_0(P),w_{\infty}=w_{\infty}(P)\in \Aut(\Sigma)$ with $w=w_0w_{\infty}$ such that $w_0\in P$ and
such that $w_{\infty}$ is $P$-reduced. 

Moreover, if $P=\Stab_W(R_x)$ for some $w$-essential point $x\in\Min(w)$, and $v\in W$ is such that $\proj_{R_x}(C_0)=vC_0$, then $v^{-1} w_{\infty}v$ is straight.
\end{prop}
\begin{proof}
Let $x\in \Min(w)$ be $w$-essential and such that $\Stab_W(R_x)=P$. Let $C:=\proj_{R_x}(C_0)$ and let $a\in P$ with $aC=\proj_{R_x}(wC_0)$. Then $\proj_{R_x}(a^{-1} wC_0)=C$, and hence $C_0$ and $a^{-1} wC_0$ lie on the same side of every wall of $