A flat plane that is not the limit of periodic\\flat planes
Daniel T. Wise
We construct a compact nonpositively curved squared $2$-com\-p\-lex whose universal cover contains a flat plane that is not the limit of periodic flat planes.
agt-3-6.tex
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%%%%     A flat plane that is not the limit of periodic flat planes
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%%%%                           Daniel T Wise 
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%%%%               Published in Volume 3(2003) pages 147-154
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%%%%                  Publication date 20 February 2003
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\title
[A flat that is not the limit of periodic flats]
{A flat plane that is not the limit of periodic\\flat planes}
\author{Daniel T. Wise}
\address{Department of Mathematics and Statistics, McGill 
University\\Montreal, Quebec H3A 2K6, Canada}
\email{wise@math.mcgill.ca}

\primaryclass{20F67} % Hyperbolic groups and nonpositively curved groups
\secondaryclass{20F06} %Cancellation theory; application of van Kampen diagrams


\keywords{CAT(0), periodic flat planes, C(4)-T(4) complexes}
\date{\today}


\begin{abstract}
We construct a compact nonpositively curved squared $2$-com\-p\-lex
whose universal cover contains a flat plane that is not the limit
of periodic flat planes.
\end{abstract}

\asciiabstract{We construct a compact nonpositively curved squared
2-complex whose universal cover contains a flat plane that is not the
limit of periodic flat planes.}


\maketitle

\section{Introduction}


Gromov raised the question of which ``semi-hyperbolic spaces''
have the property that their flats can be approximated by periodic flats
\cite[\S6.B$_3$]{Gromov93}.
%
In this note we construct an example of a
compact nonpositively curved squared $2$-complex $Z$
whose universal cover $\tilde Z$ contains an isometrically embedded
flat plane that is not the limit of a sequence of periodic flat planes.


A flat plane $\Euclidean \hookrightarrow \tilde Z$ is {\em periodic}
if the map $\Euclidean\looparrowright Z$ factors as
$\Euclidean\rightarrow T\rightarrow Z$ where $\Euclidean \rightarrow T$
is a covering map of a torus $T$.
Equivalently, $\pi_1Z$ contains a subgroup isomorphic to
 $\integers\times\integers$
which stabilizes $\Euclidean$ and acts cocompactly on it.
%
A flat plane $f\co\Euclidean\hookrightarrow \tilde Z$ is the
{\em limit of periodic flat planes}
if there is a sequence of periodic flat planes
$f_i\co\Euclidean\hookrightarrow  \tilde Z$ which converge
pointwise to $f\co\Euclidean\rightarrow \tilde Z$.
In our setting, $\tilde Z$ is a $2$-dimensional complex,
and so $\Euclidean\hookrightarrow \tilde Z$ is the limit of periodic flat planes
 if and only if every compact subcomplex of $\Euclidean$ is contained in a periodic flat plane.


In Section~\ref{sec:X antitorus} we describe a compact
nonpositively curved $2$-complex $X$ whose universal cover
contains a certain aperiodic plane called an ``anti-torus''.
In Section~\ref{sec:Z} we construct $Z$ from $X$ by strategically
gluing tori and cylinders to $X$ so that $\tilde Z$ contains
a flat plane which is a mixture of the anti-torus and periodic planes.
This flat plane is not approximable by periodic flats because
it contains a square that does not lie in any periodic flat.
%
Our example $Z$ is a $K(\pi,1)$ for a negatively
curved square of groups, and in Section~\ref{sec:polygon}
we describe an interesting related triangle of groups.





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%% THE COMPLEX X
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{The anti-torus in $X$}\label{sec:X antitorus}
\subsection{The $2$-complex $X$}
Let $X$ denote the complex consisting of the six squares
indicated in Figure~\ref{fig:sixsquares}. The squares are glued together as
indicated by the oriented labels on the edges. Note that $X$ has a unique
0-cell, and that the notion of vertical and horizontal
are preserved by the edge identifications.
Let $H$ denote the subcomplex consisting of the
$2$~horizontal edges, and let  $V$ denote  the subcomplex
consisting of the $3$~vertical edges.

\begin{figure}\centering
\includegraphics[width=\textwidth]{six.eps}
\caption{The figure above indicates the gluing pattern for the six squares of
$X$. The three vertical edges colored white, grey, and  black
are denoted $a$, $b$, and  $c$ respectively. The two horizontal
edges, single and double arrow, are denoted $x$ and $y$
respectively.\label{fig:sixsquares}}
\end{figure}


The complex $X$, which was first studied in \cite{Wise96Thesis},
has a number of interesting properties that we record here:
The link  of the unique $0$-cell in $X$ is a complete bipartite graph.
It follows that the universal cover $\widetilde X$ is the product of two trees
$\tilde H\times \tilde V$ where $\tilde H$ and $\tilde V$ are the universal covers
of $H$ and $V$.
In particular, the link contains no cycle of length~$<4$ and so
$X$ satisfies the combinatorial nonpositive curvature condition
for squared $2$-complexes \cite{Gromov87,BridsonHaefliger} which is a special
case of the $C(4)$-$T(4)$ small-cancellation condition \cite{LS77}.

The $2$-complex $X$ was used in
\cite{Wise96Thesis} to produce the first examples of non-residually finite
groups which are fundamental groups of spaces with the above properties.
%
The connection to finite index subgroups arises because
while $\widetilde X$ is isomorphic to the cartesian product of two trees,
 $X$ does not have a finite cover which is the
product of two graphs.

\subsection{The anti-torus $\Pi$}

The exotic behavior of $X$ can be attributed to
the existence of a strangely aperiodic plane $\Pi$ in $\widetilde X$
that we shall now describe.
%
Let $\tilde x\in \widetilde X^0$ be the basepoint of $\widetilde X$.
Let $c^\infty$ denote
the infinite periodic vertical line in $\tilde X$ which is the
based component of the preimage of the loop labeled by $c$ in $X$.
Define $y^\infty$ analogously.
Let $\Pi$ denote the convex hull
in $\tilde X$ of the infinite geodesics labeled by $c^\infty$ and
$y^\infty$, so $\Pi=y^\infty\times c^\infty$. The plane  $\Pi$ is tiled by the six orbits of
squares in $\tilde X$ as in Figure~\ref{fig:plane}.
The reader can extend $c^\infty\cup y^\infty$
to a flat plane by successively adding squares wherever
there is a pair of vertical and horizontal edges meeting at a vertex.
From a combinatorial point of view,
the existence and uniqueness of this extension is guaranteed by
the fact that the link of $X$ is a complete bipartite graph.

\begin{figure}\centering
\includegraphics[width=.4\textwidth]{plane.eps}
\caption{ The Anti-Torus $\Pi$: The plane $\Pi$ above is the
convex hull of two periodically labeled lines in $\tilde X$. A
small region of the northeast quadrant has been tiled by the
squares of $X$. \label{fig:plane}}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%
The ``axes'' $c^\infty$ and $y^\infty$ of $\Pi$ are obviously periodic,
and using that $X$ is compact, it is easy to verify that
for any $n\in\naturals$, the infinite strips
$[-n,n]\times \reals$ and $\reals\times [-n,n]$ are periodic.
However, the period of these infinite strips increases exponentially
with~$n$.
Thus, the entire plane $\Pi$ is aperiodic.
Note that to say that $[-n,n]\times \reals$ is {\em periodic}
means that the immersion $\big([-n,n]\times \reals\big)\looparrowright X$
factors as $\big([-n,n]\times \reals\big)\rightarrow C \looparrowright X$
where $\big([-n,n]\times \reals\big)\rightarrow C$ is the universal
covering map
of a cylinder. The map $\Pi\looparrowright X$ is {\em aperiodic} in the sense
that it does not factor through an immersed torus.


We conclude this section by giving a brief explanation
of the aperiodicity of $\Pi$.
A complete proof that $\Pi$ is aperiodic is given in \cite{Wise96Thesis}.
%
Let $W_n(m)$ denote the word corresponding to the length~$n$ horizontal
 positive path in $\Pi$  beginning at the endpoint of the vertical path $c^m$.
 Thus, $W_n(m)$ is the label of the side opposite $y^n$
in the rectangle which is the combinatorial convex hull of $y^n$ and $c^m$.
Equivalently, $W_n(m)$ occupies the interval $\{m\}\times [0,n]$.
%
For each $n$, the words
$\{W_n(m) \mid  0 \leq m \leq 2^n -1 \}$ are all distinct!
Consequently every  positive length~$n$ word  in $x$ and $y$ is $W_n(m)$ for some $m$.
This implies that the infinite strip $[0,n]\times \reals$
has period $2^n$, and in particular $\Pi$ cannot be periodic.

We refer to $\Pi$ as an {\em anti-torus} because the aperiodicity of
$\Pi$ implies that $c$ and $y$ do not have nonzero powers which commute.
Indeed, if $c^p$ and $y^q$ commuted for $p,q\neq0$ then the flat torus
theorem (see \cite{BridsonHaefliger}) would imply that
$c^\infty$ and $y^\infty$ meet in a periodic flat plane,
which would contradict that $\Pi$ is aperiodic.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%% NOT LIMIT OF PERIODIC FLATS
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{The $2$-complex $Z$ with a nonapproximable flat}
 \label{sec:Z}
 We first construct a new complex $Y$ as follows: Start with a square
$s$, and then attach four cylinders each of which is isomorphic
to $S^1\times I$. One such cylinder is attached  along each side of $s$.
The resulting complex $Y$ containing exactly five squares is illustrated in
Figure~\ref{fig:Y}.

\begin{figure}\centering
\includegraphics[width=.2\textwidth]{Y.eps}
\caption{ The complex $Y$ is formed by gluing four cylinders to a square.
\label{fig:Y}}
\end{figure}


Let $T^2$ denote the torus $S^1\times S^1$ with the usual product cell structure
consisting of one $0$-cell, two $1$-cells, and a single square $2$-cell.
We let $\widetilde T^2$ denote the universal cover and we shall identify
$\widetilde T^2$ with $\reals^2$.


At each corner of $s \subset Y$, there is a pair of intersecting circles in $Y^1$,
which are boundary circles of distinct cylinders.
Note that they meet at an angle of $\frac{3\pi}2$ in $Y$.
%
%
At each of three (NW, SW, \& SE)
 corners of $s \subset Y$ we attach a copy of $T^2$
 by identifying the pair of circles in the  $1$-skeleton of $T^2$
 with the pair of intersecting circles noted above at
the respective corner of $s$.
%
At the fourth (NE) corner of $s$, we attach a copy of the complex
$X$. Here we identify the pair of circles meeting at the corner of
$s$ with the pair of perpendicular circles $c$ and $y$ of $X$.
%
We denote the resulting complex by $Z$. Thus, $Z =   T^2 \cup T^2 \cup
T^2 \cup Y \cup X$. See Figure~\ref{fig:compandplane} for a
depiction of the $8$ squares of $Z - X$ and their gluing patterns.


\begin{figure}\centering
\includegraphics[width=.4\textwidth]{anex.eps}
\hspace{5mm}
\includegraphics[width=.5\textwidth]{badplane.eps}
\caption{
\label{fig:compandplane} $Z-X$ and $Z$: The eight squares of the
figure on the left are glued together following the gluing pattern
to form $Z-X$. To form $Z$, we add a copy of $X$ at the $NE$
corner, identifying the loops in $X$ labeled by $c$ and $y$, with
the black single and double arrows of the diagram. The figure on
the right represents an infinite cross whose convex hull in $Z$
is not approximable by any periodic plane. Note that while the NW,
SW, and SE quarters of this plane are periodic,  the NE quarter
is an aperiodic quarter of $\Pi$.}
\end{figure}

\begin{defn}[Infinite cross]
An {\em infinite cross}
is a squared $2$-complex isomorphic to the subcomplex of $\widetilde T^2$
consisting of $\big([0,1] \times \reals\big)
\cup \big(\reals \times [0,1]\big)$.
The {\em base square} of the infinite cross is the square $[0,1]\times[0,1]$.
\end{defn}


{\bf  The planes containing $s$:}\qua
Observe that $Y$ contains various immersions of an infinite
cross whose base square maps to $s$.
In particular, there are exactly 16~distinct immersed infinite crosses
$C\looparrowright Y$ that pass through $s$ exactly once.
Each of these infinite crosses extends uniquely to an immersed flat plane
in $Z$.
Each such flat plane fails to be periodic because its four quarters map to distinct
parts of $Z$.
Our main result is that these immersed flat planes
are not approximable by periodic flat planes because of the following:

\begin{thm}[No periodic approximation] There is no
immersion of a torus $T^2 \rightarrow Z$ which contains $s$.
Equivalently, there is no periodic plane in $\tilde Z$ containing
$\tilde s$.
\end{thm}
\begin{proof}
 We argue by contradiction. Suppose that there is an immersed periodic plane
$\Omega$ containing $s$.
We shall now produce a rectangle as in Figure~\ref{fig:contradiction}
that will yield a contradiction.
We may assume that a copy of $s$ in $\Omega$ is oriented
as in Figure~\ref{fig:compandplane}.
We begin at this copy of $s$ and travel
north inside the northern cylinder until we reach another
copy $s_n$ of $s$. The existence of $s_n$ is guaranteed by our assumption
that $\Omega$ is periodic.
Similarly, we travel east from $s$ to reach a square $s_e$.
Travelling north from $s_e$ and east from $s_n$, we trace out the
boundary of a rectangle whose NE corner is a  square $s_{ne}$
(see Figure~\ref{fig:contradiction}).

This yields a contradiction because the inside of this
rectangle is tiled by squares in $X$, yet the boundary of this
rectangle is a commutator $\big [ c^{\pm n}, y^{\pm m} \big] $.
As explained in Section~\ref{sec:X antitorus},
such a word cannot be trivial in $\pi_1X$
 because of the anti-torus.
\end{proof}


\begin{figure}\centering
\includegraphics[width=.5\textwidth]{contra.eps}
\caption{
The figure above
illustrates one of the four possible contradictions which explain
why no  periodic plane contains the square $s$.
\label{fig:contradiction}}
\end{figure}

\begin{rem}
Using an argument similar to the above proof, one can show
that these sixteen planes are the {\it only} flat planes in $\tilde Z$ containing
$\tilde s$.
One considers the pair of ``axes''
intersecting at $\tilde s$ in a plane containing $\tilde s$. If this plane is
different from each of the $16$ mentioned above, then some translate
of $\tilde s$ must appear along one of these ``axes''. The infinite strip
in the plane whose corners are these two $s$ squares
yields a contradiction similar to the one obtained above.
\end{rem}

\begin{rem}
While $X$ is a rather pathological complex,
we note that every flat plane in $\tilde X$ is the limit of periodic flat planes.
Indeed this holds for any compact $2$-complex $X$ whose universal cover
is isomorphic to the product of two trees
\cite{Wise96Thesis}.
\end{rem}

\section{Polygons of groups}
\label{sec:polygon}
\subsection{The algebraic angle versus the geometric angle}
Since the elements $c$ and $y$ have axes which
intersect perpendicularly in a plane in $\tilde X$, the
natural geometric angle between the subgroups $\langle c\rangle$ and
$\langle y\rangle$ of $\pi_1X$ is $\frac\pi2$.
However, the algebraic Gersten-Stallings angle (see \cite{Stallings91}) between
these subgroups is $\leq \frac\pi3$.
%
To see this, we must show that there is no non-trivial
relation of the form $c^k y^l c^m y^n = 1$.

Since $\tilde X$ is isomorphic to the cartesian product $\tilde V\times \tilde H$,
of two trees and $c$ and $y$ correspond to distinct factors,
it follows that the only relations that must be checked are
rectangular
 (i.e., $\vert k\vert = \vert m\vert$ and
        $\vert l\vert = \vert n\vert$ ).
        However, these are
easily ruled out by the anti-torus $\Pi$ and the fact that $X$ is nonpositively curved.

\subsection{Square of groups and triangle of groups}
The complex $Z$ can be  thought of in a natural way
as a $K(\pi,1)$ for a negatively curved {\it square of groups}
(see \cite{Stallings91,Haefliger91,Corson92}) with cyclic edge groups
and trivial face group.

Because  the algebraic angle between $\langle c \rangle$ and
$\langle y \rangle$ in $\pi_1X$ is $\leq \frac\pi3$, it is tempting to form an analogous
nonpositively curved triangle of groups $D$.
The face group of $D$ is trivial, the edge groups
of $D$ are cyclic, the vertex groups of $D$ are isomorphic to $\pi_1X$, and
each edge group of $D$ is embedded on one (clockwise) side as
$\langle c \rangle$ and on the other (counter-clockwise) side as
$\langle y \rangle$. This can be done so that the resulting
triangle of groups $D$ has $\integers_3$ symmetry.
%
The tension between the algebraic and geometric angles
should endow $\pi_1D$ with some interesting properties.
For instance, I suspect that $\pi_1D$ fails to be the fundamental group
of a compact nonpositively curved space, but it fails for reasons different
from the usual types of problems.

{\bf Acknowledgments}\qua I am grateful to the referee for the helpful
corrections.  This research was supported by an NSERC grant.


\begin{thebibliography}

\bibitem{BridsonHaefliger}
\textbf{Martin~R Bridson}, \textbf{Andr{\'e} Haefliger}, \emph{Metric spaces of
  non-positive curvature}, Springer-Verlag, Berlin (1999)

\bibitem{Corson92}
\textbf{Jon~Michael Corson}, \emph{Complexes of groups}, Proc. London Math.
  Soc. (3) 65 (1992) 199--224

\bibitem{Gromov87}
\textbf{M Gromov}, \emph{Hyperbolic groups}, from: ``Essays in group theory'',
  Math. Sci. Res. Inst. Publ. 8, Springer, New York (1987)  75--263

\bibitem{Gromov93}
\textbf{M Gromov}, \emph{Asymptotic invariants of infinite groups}, from:
  ``Geometric group theory, Vol.\ 2 (Sussex, 1991)'', Cambridge Univ. Press,
  Cambridge (1993)  1--295

\bibitem{Haefliger91}
\textbf{Andr{\'e} Haefliger}, \emph{Complexes of groups and orbihedra}, from:
  ``Group theory from a geometrical viewpoint (Trieste, 1990)'', ({\'E} Ghys, A
  Haefliger, A Verjovsky, editors), World Sci. Publishing, River Edge, NJ
  (1991)  504--540

\bibitem{LS77}
\textbf{Roger~C Lyndon}, \textbf{Paul~E Schupp}, \emph{Combinatorial group
  theory}, Springer-Verlag, Berlin (1977), ergebnisse der Mathematik und ihrer
  Grenz\-gebiete, Band 89

\bibitem{Stallings91}
\textbf{John~R Stallings}, \emph{Non-positively curved triangles of groups},
  from: ``Group theory from a geometrical viewpoint (Trieste, 1990)'', ({\'E}
  Ghys, A Haefliger, A Verjovsky, editors), World Sci. Publishing, River Edge,
  NJ (1991)  491--503

\bibitem{Wise96Thesis}
\textbf{Daniel~T Wise}, \emph{Non-positively curved squared complexes,
  aperiodic tilings, and non-residually finite groups}, Ph.D. thesis, Princeton
  University (1996)

\end{thebibliography}




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math0303048.extracted_bib
\begin{thebibliography}

\bibitem{BridsonHaefliger}
\textbf{Martin~R Bridson}, \textbf{Andr{\'e} Haefliger}, \emph{Metric spaces of
  non-positive curvature}, Springer-Verlag, Berlin (1999)

\bibitem{Corson92}
\textbf{Jon~Michael Corson}, \emph{Complexes of groups}, Proc. London Math.
  Soc. (3) 65 (1992) 199--224

\bibitem{Gromov87}
\textbf{M Gromov}, \emph{Hyperbolic groups}, from: ``Essays in group theory'',
  Math. Sci. Res. Inst. Publ. 8, Springer, New York (1987)  75--263

\bibitem{Gromov93}
\textbf{M Gromov}, \emph{Asymptotic invariants of infinite groups}, from:
  ``Geometric group theory, Vol.\ 2 (Sussex, 1991)'', Cambridge Univ. Press,
  Cambridge (1993)  1--295

\bibitem{Haefliger91}
\textbf{Andr{\'e} Haefliger}, \emph{Complexes of groups and orbihedra}, from:
  ``Group theory from a geometrical viewpoint (Trieste, 1990)'', ({\'E} Ghys, A
  Haefliger, A Verjovsky, editors), World Sci. Publishing, River Edge, NJ
  (1991)  504--540

\bibitem{LS77}
\textbf{Roger~C Lyndon}, \textbf{Paul~E Schupp}, \emph{Combinatorial group
  theory}, Springer-Verlag, Berlin (1977), ergebnisse der Mathematik und ihrer
  Grenz\-gebiete, Band 89

\bibitem{Stallings91}
\textbf{John~R Stallings}, \emph{Non-positively curved triangles of groups},
  from: ``Group theory from a geometrical viewpoint (Trieste, 1990)'', ({\'E}
  Ghys, A Haefliger, A Verjovsky, editors), World Sci. Publishing, River Edge,
  NJ (1991)  491--503

\bibitem{Wise96Thesis}
\textbf{Daniel~T Wise}, \emph{Non-positively curved squared complexes,
  aperiodic tilings, and non-residually finite groups}, Ph.D. thesis, Princeton
  University (1996)

\end{thebibliography}