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\controldates{9-DEC-1998,9-DEC-1998,9-DEC-1998,9-DEC-1998}
 
\documentclass{era-l}

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\begin{document}

\title{On aspherical presentations of groups}
\author{Sergei V. Ivanov}
\address{Department of Mathematics, 
University of Illinois at Urbana-Champaign, 
1409 West Green Street,   Urbana, 
IL 61801}
\email{ivanov@math.uiuc.edu}
\subjclass{Primary 20F05, 20F06, 20F32; Secondary 57M20}
\thanks{Supported in part by an Alfred P. Sloan Research
Fellowship and  NSF grant DMS 95-01056}
\issueinfo{4}{15}{}{1998}
\dateposted{December 15, 1998}
\pagespan{109}{114}
\PII{S 1079-6762(98)00052-3}
\def\copyrightyear{1998}
\copyrightinfo{1998}{American Mathematical Society}
\commby{Efim Zelmanov}
\date{April 13, 1998}
\begin{abstract}The Whitehead asphericity conjecture claims 
that if $\langle \, \mathcal{A} \, \| \,  \mathcal{R} \, \rangle $ is  
an 
aspherical
group presentation, then for every $\mathcal{S} \subset \mathcal{R}$ the 
subpresentation $\langle \, \mathcal{A} \, \| \,  \mathcal{S} \, 
\rangle $ is also
aspherical. This conjecture is generalized for presentations of groups 
with 
periodic elements by introduction of almost aspherical presentations.
It is  proven that the generalized Whitehead asphericity conjecture
(which claims that every subpresentation of an almost aspherical
presentation is also almost aspherical)
is equivalent to the original Whitehead conjecture and
holds for standard presentations of free Burnside groups of large odd 
exponent,
Tarski monsters and some others. 
Next, it is proven that 
if the Whitehead  conjecture is false, then there is an aspherical 
presentation 
$E = \langle \, \mathcal{A} \, \| \,  \mathcal{R} \cup z \, \rangle $ 
of the 
trivial group $E$, where the alphabet 
$\mathcal{A}$ is finite or countably infinite and $z \in \mathcal{A}$,  
such that 
 its subpresentation
$\langle \, \mathcal{A} \, \| \,  \mathcal{R} \, \rangle $ is not 
aspherical. It 
is also
proven that if 
the Whitehead conjecture fails for finite presentations (i.e., with
finite $\mathcal{A}$ and $\mathcal{R}$), then there is a finite 
aspherical 
presentation $\langle \, \mathcal{A} \, \| \,  \mathcal{R} \, \rangle $, 
$\mathcal{R} = \{ R_{1}, R_{2}, \dots , R_{n} \}$,  such that for every 
$\mathcal{S} \subseteq \mathcal{R}$ 
the subpresentation $\langle \, \mathcal{A} \, \| \,  \mathcal{S} \, 
\rangle $ 
is aspherical and the subpresentation
 $\langle \, \mathcal{A} \, \| \,  R_{1}R_{2}, R_{3}, \dots , R_{n}\, 
\rangle $ of  
aspherical
$\langle \, \mathcal{A} \, \| \,  R_{1}R_{2}, R_{2}, R_{3}, \dots , 
R_{n}\, \rangle $ 
is not aspherical.
Now suppose  a group presentation $H = \langle \, \mathcal{A} \, \| \,  
\mathcal{R} \, \rangle $
is aspherical, $x \not \in \mathcal{A}$, $W(\mathcal{A} \cup x)$ is a 
word in the
alphabet
$(\mathcal{A} \cup x)^{\pm 1}$ with nonzero sum of exponents on $x$, 
and the 
group
$H$ naturally embeds in 
$G = \langle \, \mathcal{A} \cup x \, \| \,  \mathcal{R}  \cup 
W(\mathcal{A} \cup x) 
\, \rangle $.
It is conjectured that the presentation 
$G = \langle \, \mathcal{A} \cup x \, \| \,  \mathcal{R}  \cup 
W(\mathcal{A} \cup x) 
\, \rangle $
is  aspherical  if and only if $G$ is torsion free. It is proven that
if this conjecture is false and 
$G = \langle \, \mathcal{A} \cup x \, \| \,  \mathcal{R}  \cup 
W(\mathcal{A} \cup x) \,
\rangle $ is a counterexample, then the integral group ring 
$\mathbb{Z}(G)$ 
of the
torsion free group $G$ will contain zero divisors. Some special cases
where this conjecture holds are also indicated.

\end{abstract}
\maketitle



Let  
\begin{equation}\label{eq:1}
\langle \, \mathcal{A} \, \| \,  \mathcal{R} \, 
\rangle
\end{equation}  
be a group presentation, where $\mathcal{A}$ is an alphabet, 
and $\mathcal{R}$ is a
set of  defining relators (which are words in 
$\mathcal{A}^{\pm 1}=\mathcal{A} \cup \mathcal{A}^{-1}$).
The group $G$ given by this presentation is the quotient 
$F(\mathcal{A})/N(\mathcal{R})$, where
$F(\mathcal{A})$ is the free group over the alphabet $\mathcal{A}$ 
and $N(\mathcal{R})$ is the normal closure of  $\mathcal{R}$ in 
$F(\mathcal{A})$. The 
quotient
$N(\mathcal{R})/N'(\mathcal{R})$, where $N'(\mathcal{R})$ is the 
commutator subgroup of 
$N(\mathcal{R})$,  can be turned into a (left)  $G$-module as follows: 
If $\mya : F(\mathcal{A}) \to G$ and $\myb : N(\mathcal{R}) \to 
N(\mathcal{R})/N'(\mathcal{R})$ are natural epimorphisms, and $W \in 
F(\mathcal{A})$, $S \in N(\mathcal{R})$, 
then
\begin{equation*}W^{\mya }\cdot S^{\myb }= (WSW^{-1})^{\myb }. 
\end{equation*}
Clearly, this $G$-action extends to an action 
of the integral group ring $\mathbb{Z}(G)$ of $G$ over $\mathcal{M}(G) 
= N(\mathcal{R})/N'(\mathcal{R})$ by setting
\begin{equation*}(W_{1}^{\mya }\pm W_{2}^{\mya }) \cdot S^{\myb }= 
(W_{1}SW_{1}^{-1}W_{2}S^{\pm 1}W_{2}^{-1})^{\myb }. 
\end{equation*}
This $\mathbb{Z}(G)$-module $\mathcal{M}(G)$ is called the {\em 
relation module} 
of 
 $G = \langle \, \mathcal{A} \, \| \,  \mathcal{R} \, \rangle $. A group 
presentation \eqref{eq:1}  is
called  {\em aspherical} if  its relation module 
is freely generated by the images $R^{\myb }$ of relators $R \in 
\mathcal{R}$. If 
$K_{G}$ is a 
2-complex associated with $G = \langle \, \mathcal{A} \, \| \,  
\mathcal{R} \, 
\rangle $ in a standard way
($K_{G}$ has a single 0-cell and $\pi _{1}(K_{G})=G$), then $G$ is 
aspherical 
if and only if 
so is $K_{G}$ (see \cite{GR}; we recall that a 2-complex $K$ is called 
aspherical 
if $\pi _{2}(K)=0$). 

The Whitehead asphericity conjecture (originally stated as a question
in \cite{W}) claims that every subcomplex of an aspherical 2-complex $K$ 
is also aspherical. The problem has received a great deal of attention 
(see \cite{GR}, \cite{H1}, \cite{H2}, \cite{Hb2}, \cite{Lf} and 
references there) but still is far from being solved.
A remarkable result on this conjecture proven by Howie \cite{H1}
reduces  the conjecture to asphericity of subcomplexes of  aspherical 
contractible
2-complexes. More specifically, it was shown in \cite{H1} that 
if the Whitehead conjecture is false, then there is a counterexample of 
one of the
following two types:

\begin{itemize}
\item[1.] $K$ is a finite aspherical contractible 2-complex and $L$ is
nonaspherical subcomplex of $K$ obtained from $K$ by removing one 
2-cell.

\item[2.]  $K$ is an aspherical contractible 2-complex, 
$K=\bigcup _{i=1}^{\infty }L_{i}$, \ $L_{i} \subset L_{i+1}$, the inclusion 
$L_{i} \to L_{i+1}$ is nullhomotopic, and each $L_{i}$ is
finite and is not aspherical.
\end{itemize}

Recently Luft \cite{Lf} reproved  Howie's result and showed that the
existence of a counterexample of type 1 implies the
existence of a counterexample of type 2.

Clearly, in group-theoretic terms the Whitehead conjecture is rephrased 
as follows: 
If presentation \eqref{eq:1} is aspherical, then every subpresentation of \eqref{eq:1} of 
the
form $\langle \, \mathcal{A} \, \| \,  \mathcal{R}' \, \rangle $ with 
$\mathcal{R}' 
\subset \mathcal{R}$
is also aspherical. 

First we will see that it is possible to assume in the Whitehead 
asphericity conjecture that the removed part $\mathcal{R} \setminus 
\mathcal{R}'$ 
of $\mathcal{R}$ 
is just a letter of $\mathcal{A}$. We recall that a presentation is called {\em 
finite} if both  $\mathcal{A}$ and $\mathcal{R}$ are finite.

\begin{thm}\label{thm:1} If the Whitehead asphericity conjecture is false,
then there is an aspherical presentation 
$E = \langle \, \mathcal{A} \, \| \,  \mathcal{R} \cup z \, \rangle $ 
of the 
trivial group $E$, where the alphabet 
$\mathcal{A}$ is finite or countably infinite and $z \in \mathcal{A}$,  
such that 
 its subpresentation
$\langle \, \mathcal{A} \, \| \,  \mathcal{R} \, \rangle $ is not 
aspherical. In 
addition, if there is a finite presentation giving a counterexample to 
the Whitehead asphericity conjecture, then there is a finite 
presentation 
$\langle \, \mathcal{A} \, \| \,  \mathcal{R} \cup z \, \rangle $ such 
that its 
subpresentation
$\langle \, \mathcal{A} \, \| \,  \mathcal{R} \, \rangle $ is not 
aspherical. 
\end{thm}


We recall that elementary Andrews-Curtis {\em transformations} over a finite 
group presentation 
$\langle \, \mathcal{A} \, \| \,  \mathcal{R} \, \rangle $  of types 
(T1)--(T3) 
are defined as follows:

\begin{itemize}
\item[(T1)] Add a new letter $b \not \in \mathcal{A}$ to both  
$\mathcal{A}$ and   
$\mathcal{R}$.

\item[(T2)] If $a \in \mathcal{A}$,  $a \in \mathcal{R}$, and 
$a, a^{-1}$ do not occur in relators $R \in \mathcal{R} \setminus a $,
then delete
$a$ in both $\mathcal{A}$ and   $\mathcal{R}$.

\item[(T3)] Replace $R \in \mathcal{R}$ by $C_{1} R^{\mye }C_{1}^{-1}  
C_{2} S^{\myd }C_{2}^{-1}$, where
$\mye , \myd = \pm 1$, $C_{1}, C_{2} \in F(\mathcal{A})$, and $S \in 
\mathcal{R} \setminus R$.
\end{itemize}

Two finite presentations are called  Andrews-Curtis {\em equivalent} 
if one of them can be obtained from the other by a finite sequence of 
elementary Andrews-Curtis transformations. (We recall that another major
problem of low-dimensional topology, the so-called Andrews-Curtis
conjecture \cite{AC}, asks whether a finite aspherical presentation of 
the trivial group is Andrews-Curtis equivalent to 
$\langle \, \mathcal{A} \, \| \, \mathcal{A} \, \rangle $.)

Clearly,  transformations (T1)--(T3) preserve the asphericity of a 
presentation  
$\langle \, \mathcal{A} \, \| \,  \mathcal{R} \, \rangle $. Moreover, 
(T1)--(T2) 
evidently preserve 
the asphericity of subpresentations. Whether (T3) preserves the 
asphericity of subpresentations is 
unclear and turns out to be  equivalent to the Whitehead asphericity 
conjecture for finite presentations following from

\begin{thm}\label{thm:2} Suppose $\langle \, \mathcal{A} \, \| \,  \mathcal{R} 
\, 
\rangle $ is a finite aspherical
presentation. Then $\langle \, \mathcal{A} \, \| \,  \mathcal{R} \, 
\rangle $ is 
Andrews-Curtis equivalent 
(with a single (T1),  no (T2) and several (T3)'s) to a finite 
aspherical presentation 
$\langle \, \mathcal{B} \, \| \,  \mathcal{S} \, \rangle $ such that 
for every 
$\mathcal{S}' \subseteq \mathcal{S}$
the subpresentation $\langle \, \mathcal{B} \, \| \,  \mathcal{S}' \, 
\rangle $ 
is aspherical. 
\end{thm}


For 2-complexes (for definitions see \cite{S} or \cite{H2}) Theorem \ref{thm:2} 
implies 

\begin{theorem3} Every finite aspherical $2$-complex can be 
$3$-deformed to a 
finite $2$-com\-plex all of whose subcomplexes are aspherical.
\end{theorem3}


Technical details  in proving Theorem \ref{thm:2} enable us to sharpen the 
equivalence
between the Whitehead asphericity conjecture for finite presentations 
and preservation of asphericity of subpresentations under
(T3) as follows:

\begin{thm}  If the Whitehead asphericity conjecture is false 
for finite presentations,
then there is a finite aspherical presentation $\langle \, \mathcal{A} 
\, 
\| \,  \mathcal{R} \, \rangle $, 
$\mathcal{R} = \{ R_{1}, R_{2}, \dots , R_{n} \}$,  such that for every 
$\mathcal{S} \subseteq \mathcal{R}$ 
the subpresentation $\langle \, \mathcal{A} \, \| \,  \mathcal{S} \, 
\rangle $ 
is aspherical and the sub\-presentation
 $\langle \, \mathcal{A} \, \| \,  R_{1}R_{2}, R_{3}, \dots , R_{n}\, 
\rangle $ of  
aspherical
$\langle \, \mathcal{A} \, \| \,  R_{1}R_{2}, R_{2}, R_{3}, \dots , 
R_{n}\, \rangle $ 
is not aspherical.
\end{thm}


It is well known 
and easy to show  that if \eqref{eq:1} is aspherical, then none of $R \in 
\mathcal{R}$ is a
proper power in the free group $F(\mathcal{A})$ over $\mathcal{A}$ and 
the group
$G$ is torsion free. In particular, presentations of periodic groups
(like those of free Burnside groups, Tarski monsters and others 
constructed in
\cite{O1}--\cite{O3}, \cite{IO1}, \cite{I}, \cite{IO2}) seem to have 
nothing to do with 
aspherical presentations and the Whitehead conjecture. This, however, 
is not the case and we will see that the Whitehead conjecture has a 
natural expansion to groups with periodic elements. More specifically, 
let a group $G$ be given by  presentation  \eqref{eq:1}.
For every $R \in \mathcal{R}$ we  let $R = Q_{R}^{m_{R}}$ in the free 
group 
$F(\mathcal{A})$, where $Q_{R}$ is not proper in $F(\mathcal{A})$ and 
$m_{R} \ge 1$. Following \cite{GR}, we call presentation \eqref{eq:1}  {\em 
almost 
aspherical} if defining relations of  $\mathcal{M}(G)$ (with generating 
set 
$\{ R^{\myb }| R \in \mathcal{R} \}$) look like
\begin{equation*}(1-Q_{R}^{\mya })\cdot R^{\myb }= 0, \ \ R \in 
\mathcal{R}
\end{equation*}
(note that if none of $R$ is a proper power, that is, $m_{R}$ are all 1, 
then we have the foregoing definition of an aspherical presentation for 
 $1-Q_{R}^{\mya }= 1-R^{\mya }=0$ in the integral group ring 
$\mathbb{Z}(G)$ of $G$).

For example, using this definition, one can restate the main result of 
Lyndon's article \cite{Ln} as follows: A one-relator group presentation 
is almost aspherical. 
Another  interesting example of an almost aspherical presentation is 
constructed by induction on $i\ge 0$ as follows: Let $\mathcal{A}^{\pm 
1}= 
\{a_{1}^{\pm 1}, \dots , a_{m}^{\pm 1} \}$ and $F_{m} = F(\mathcal{A})$ 
be the 
free group of rank $m$ over 
$\mathcal{A}$. Following Ol'shanskii
\cite{O1} (and \cite{I}), put $B(m,n,0)= F_{m}$. Assuming that 
$B(m,n,i)$, $i \ge 0$, is already constructed as a quotient group of 
$F_{m}$, define $A_{i+1}$ to be a shortest word over $\mathcal{A}^{\pm 
1}$ (if 
any) whose image in $B(m,n,i)$ has infinite order. Then 
$B(m,n,i+1)$ is the quotient group of $B(m,n,i)$ by the relation $A_{i+
1}^{n}=1$. Clearly,
\begin{equation*}B(m,n,i+1)=  \langle \, \mathcal{A} \, \| \, 
A_{1}^{n}, A_{2}^{n},  \dots , A_{i+
1}^{n} \, \rangle \end{equation*}
provided the word $A_{i+1}$ exists (otherwise, $B(m,n,i)$ is periodic
and the construction stalls). It is shown in \cite{O1} that if $m>1$,
$n$ is odd and $n>10^{10}$, then $A_{i+1}$  always exists and the group
\begin{equation}\label{eq:2}
B(m,n,\infty )=  \langle \, \mathcal{A} \, \| \, 
A_{1}^{n}, A_{2}^{n},  \dots , A_{i+
1}^{n}, \dots \, \rangle 
\end{equation}
obtained by imposing on $F_{m}$ of all defining relations $A_{i}^{n}=1$ 
for 
$i=1,2,\dots $
is naturally isomorphic to the free Burnside group $B(m,n)$ with $m$ 
generators
and exponent $n$ (recall  that $B(m,n)$ is the quotient $F_{m} / F_{m}^{n}$,
where
$F_{m}^{n}$ is the subgroup of $F_{m}$
generated by all $n$th powers; the same results are proven in \cite{I} 
for $m >1$ and all $n\ge 2^{48}$ such that if $n$ is even, then $n$ is 
divisible by $2^{9}$). 

Moreover, it is shown by Ashmanov and
Ol'shanskii \cite{AO} (see also Chapter 10 in \cite{O3}) that \eqref{eq:2} is an 
almost aspherical presentation provided 
$m>1$, $n>10^{10}$ is odd (for even $n$ this is not the case).
It is worth mentioning  that Ol'shanskii's presentations of Tarski 
monsters \cite{O2} (see also \cite{O3}; we recall that a Tarski monster is an 
infinite group all of whose proper subgroups are cyclic of the same 
prime order $p$) are also
almost aspherical. Hence, in analogy with the Whitehead conjecture, one 
might wonder if all subpresentations of  \eqref{eq:2}  (or  of Tarski monster's 
given in \cite{O2}) are also almost aspherical. More generally, one 
could generalize the Whitehead asphericity conjecture as follows. 

\begin{theorem5}  Every subpresentation of an almost aspherical 
group
presentation is also almost aspherical. 
\end{theorem5}


Surprisingly, this turns out not to be any more general.

\begin{thm}\label{thm:4} This generalized Whitehead asphericity conjecture 
is equivalent to the original Whitehead asphericity conjecture.
\end{thm}


As a matter of fact, when proving Theorem \ref{thm:4}, we make reduction to an 
aspherical
presentation of the trivial group (similar to reductions in \cite{H1}, 
\cite{Lf}) which yields the following.

\begin{thm}\label{thm:5} If the generalized Whitehead asphericity 
conjecture is false, then there exists a counterexample  $\langle \, 
\mathcal{A} \, \| \, \mathcal{R} \, \rangle $ to the Whitehead 
asphericity 
conjecture such that  
$\langle \, \mathcal{A} \, \| \, \mathcal{R} \, \rangle $ is an 
aspherical 
presentation of the trivial group, where $\mathcal{A}$ is finite
or countably infinite and $\mathcal{R}$ contains a finite subset 
$\mathcal{R}'$ 
so that the subpresentation
 $\langle \, \mathcal{A} \, \| \, \mathcal{R}' \, \rangle $ is not 
aspherical.
\end{thm}


In view of  Theorems \ref{thm:4}--\ref{thm:5}, 
the problem whether the generalized
Whitehead asphericity conjecture holds for presentations \eqref{eq:2}
(and other group presentations of  \cite{O3}, \cite{IO1}) becomes 
especially interesting. A positive solution to this problem is provided 
by

\begin{thm} $(a)$ Let  $m>1$, $n$ odd, $n > 10^{10}$. Then 
every subpresentation
of the free Burnside group $B(m,n)$  presented by \eqref{eq:2} is almost 
aspherical.

$(b)$ Let $G = \langle \, \mathcal{A} \, \| \, \mathcal{R} \, \rangle $ 
be a 
(graded)
presentation constructed as in Chapters 6 and 8 of \cite{O3}. Then every 
subpresentation of  $\langle \, \mathcal{A} \, \| \, \mathcal{R} \, 
\rangle $ is 
almost aspherical.
In particular, if $p$ is prime, $p \gg 1$, then every 
subpresentation of Tarski monster presentation constructed in \cite{O2} 
is almost aspherical.
\end{thm}


Now consider a special type of  group presentations:
\begin{equation}\label{eq:3}G = \langle \, \mathcal{A} \cup x \, \| \,  
\mathcal{R}(\mathcal{A}) \cup W(\mathcal{A} 
\cup x) \, \rangle , 
\end{equation}
where $x \not \in \mathcal{A}$, all relators in  
$\mathcal{R}(\mathcal{A})$ are words in 
$\mathcal{A}^{\pm 1}$ and $W(\mathcal{A} \cup x)$ is a word in 
$(\mathcal{A} \cup x)^{\pm 1}$ with nonzero sum of exponents on $x$.
Assuming that \eqref{eq:3} is aspherical, one can easily reduce the problem
on asphericity of  subpresentation $\langle \, \mathcal{A} \, \| \,  
\mathcal{R} 
\, \rangle $ to whether or not  the group
$H = \langle \, \mathcal{A} \, \| \,  \mathcal{R} \, \rangle $  embeds 
in the 
group $G$ given by \eqref{eq:3}, that is,
whether the equation $W(\mathcal{A} \cup x) =1$ is solvable over $H$. 
This, 
however, is another difficult open problem (the so-called Kervaire problem 
about the solvability of equations over groups) and the affirmative 
solution is known only in some special cases. For example: If all 
occurrences of $x^{\pm 1}$ in $W$ have positive (or negative) 
exponents (Levin \cite{Lv});
If the sum of exponents on $x$ in $W$ is $\pm 1$ and $H$ is torsion
free (Klyachko \cite{K}); If $H$  is locally indicable, that is, every
nontrivial finitely generated subgroup of $H$ has an infinite cyclic
epimorphic image (Howie \cite{H3}; see also Brodskii \cite{B}).

The following  seems worth mentioning and is immediate from
the foregoing reduction and Klyachko's result.

\begin{theorem9} If \eqref{eq:3} is a balanced
presentation of the trivial group (and hence aspherical), 
then its subpresentation 
$H = \langle \, \mathcal{A} \, \| \,  \mathcal{R}(A) \, \rangle $ is 
aspherical 
if and only if $H$ is a torsion free group.
\end{theorem9}


Now let us turn tables around to indicate an interesting connection 
between
the asphericity of  presentation \eqref{eq:3}, torsion in the group $G$, and the
Kaplansky problem on
zero divisors which asks whether the group ring of a torsion 
free group  over an integral domain  can have zero divisors.

First let us state a conjecture that, like the Whitehead
asphericity conjecture, is actually a problem more convenient to state
in the affirmative form.

\begin{theorem5}\label{conj:2} Suppose  a group presentation 
$H = \langle \, \mathcal{A} \, \| \,  \mathcal{R} \, \rangle $
is aspherical, $x \not \in \mathcal{A}$, $W(\mathcal{A} \cup x)$ is a 
word in 
$(\mathcal{A} \cup x)^{\pm 1}$ with nonzero sum of exponents on $x$, 
and the 
group
$H$ naturally embeds in 
\begin{equation}\label{eq:4}
G = \langle \, \mathcal{A} \cup x \, \| \,  
\mathcal{R}  \cup W(\mathcal{A} \cup x) 
\, \rangle . 
\end{equation}
Then  presentation \eqref{eq:4} is aspherical if and only if  the group  $G$ 
is torsion free.
\end{theorem5}


\begin{thm}\label{thm:7} If Conjecture \ref{conj:2}   fails and 
$G = \langle \, \mathcal{A} \cup x \, \| \,  \mathcal{R}  \cup 
W(\mathcal{A} \cup x) 
\, \rangle $ is a counterexample
to it, then the group $G$ is a torsion free group whose integral group 
ring $\mathbb{Z}(G)$
contains zero divisors. In addition, if $W(\mathcal{A} \cup x)$ has $n$ 
occurrences of $x^{\pm 1}$,  then $\mathbb{Z}(G)$ contains a zero 
divisor 
$Z$ with $|\op {supp}Z| \le n$.
\end{thm}


Some special cases where Conjecture \ref{conj:2} holds are indicated in 

\begin{thm}\label{thm:8} Conjecture \ref{conj:2} holds in the following cases:
\begin{itemize}
\item[(a)]  The group $H = \langle \, \mathcal{A} \, \| \,  \mathcal{R} 
 \, 
\rangle $ is locally indicable.
\item[(b)]  The sum of exponents on $x$ in 
$W(\mathcal{A} \cup x)$ is $\pm 1$.
\item[(c)]   The number of  occurrences of $x^{\pm 1}$ in $W(\mathcal{A} 
\cup x)$ is 
at most $3$.
\item[(d)]  If  $\,W(\mathcal{A} \cup x) \equiv U_{1} x^{\mye _{1}} \dots 
U_{n} 
x^{\mye _{n}}$, 
where $\mye _{1}, \dots , \mye _{n} \in \{\pm 1\}$, $U_{1}, \dots , 
U_{n}$ are words in 
$\mathcal{A}^{\pm 1}$, then there are precisely two alternations of 
sign in 
the cyclic
sequence $(\mye _{1}, \dots , \mye _{n})$ and if $\mye _{k}\mye _{k+1} 
= \mye _{\ell }\mye _{\ell +
1} =-1$
(subscripts $\pmod n$) 
with $k \neq \ell $, then $U_{k+1} \neq 1$,  $U_{\ell +1} \neq 1$ in 
$H= \langle \, \mathcal{A} \, \| \,  \mathcal{R}  \, \rangle $. 
\end{itemize}
\end{thm}


Proofs of Theorems \ref{thm:1}--\ref{thm:8} make use of (more or less) 
standard techniques 
of  group theory such as Nielsen reduced bases for 
subgroups of free groups,
Reidemeister-Schreier rewriting process, Fox's derivatives, small
cancellation theory, Ol'shanskii's machinery of graded diagrams and
graded group presentations,  and van Kampen diagrams on orientable
surfaces.


\bibliographystyle{amsalpha}
\begin{thebibliography}{APS2} 



\bibitem[AO]{AO}
I.S. Ashmanov and A.Yu. Ol'shanskii, {\em On abelian and central 
extensions of aspherical groups}, Izv. Vyssh. Uchebn. Zaved. Mat.
{\bf 11} (1985), 48--60. 
\MR{87m:20095}


\bibitem[AC]{AC}
J.J. Andrews and M.L. Curtis, {\em Free groups and handlebodies}, Proc. 
Amer. Math. Soc. {\bf 16} (1965), 192--195.
 \MR{30:3454}


\bibitem[B]{B}
S.D. Brodskii, {\em Equations over groups and groups with
a single defining relation}, Uspekhi Mat. Nauk {\bf 35} (1980), 183.
 \MR{82a:20041}


\bibitem[GR]{GR}
M. Gutierrez and J.G. Ratcliffe, {\em On the second homotopy group}, 
Quart. J. Math. Oxford {\bf 32} (1981), 45--55.
 \MR{82g:57003}


\bibitem[H1]{H1}
J. Howie, {\em Some remarks on a problem of  J.H.C. Whitehead}, 
Topology {\bf 22} (1983), 475--485.
 \MR{85g:57003}


\bibitem[H2]{H2}
J. Howie, {\em On the asphericity of  ribbon disc complements}, Trans. 
Amer. Math. Soc. {\bf 289} (1985), 281--302.
 87a:57007


\bibitem[H3]{H3}
J. Howie, {\em On locally indicable groups}, Math. Z. {\bf 180} (1982), 
445--461.
 \MR{84b:20036}


\bibitem[Hb1]{Hb1}
J. Huebschmann, {\em Cohomology theory of aspherical groups and of 
small cancellation 
groups}, J. Pure Appl. Algebra {\bf 14} (1979), 137--143.
 \MR{80e:20064}


\bibitem[Hb2]{Hb2}
J. Huebschmann, {\em Aspherical 2-complexes and an unsettled problem of 
J.H.C. 
Whitehead}, Math. Ann. {\bf 258} (1981), 17--37.
 \MR{83e:57004}


\bibitem[I]{I}
S.V. Ivanov, {\em The free Burnside groups of sufficiently large 
exponents}, Internat. J. Algebra Comp. {\bf 4} (1994), 1-308.
 \MR{95h:20051}


\bibitem[IO1]{IO1}
S.V. Ivanov and A.Yu. Ol'shanskii, {\em Some applications of graded 
diagrams in combinatorial group theory}, London Math. Soc. Lecture Note 
Ser. {\bf 160} (1991), 258-308.
 \MR{92j:20022}


\bibitem[IO2]{IO2}
S.V. Ivanov and A.Yu. Ol'shanskii, {\em Hyperbolic  groups and their  
quotients of  bounded exponents}, Trans. of the Amer. Math. Soc {\bf 
348} (1996), 2091--2138.
 \MR{96m:20057}


\bibitem[K]{K}
A. Klyachko, {\em A funny property of sphere and equations over 
groups}, Comm. Algebra {\bf 122} (1994), 1475--1488.
 \MR{94c:20070}


\bibitem[Lv]{Lv}
F. Levin, {\em Solutions of equations over groups}, Bull. Amer. Math.  
Soc. {\bf 62} (1962), 603--604.
\MR{26:212}


\bibitem[Lf]{Lf}
E. Luft, {\em On 2-dimensional aspherical complexes and a problem of  
J.H.C. 
Whitehead}, Math. Proc. Cambridge Phil. Soc. {\bf 119} (1996), 493--495.
 \MR{96h:57003}


\bibitem[Ln]{Ln}
R.C. Lyndon, {\em Cohomology theory of groups with a single defining 
relation}, Ann. Math. {\bf 52} (1950), 650--655.
 \MR{13:819b}


\bibitem[LS]{LS}
R.C. Lyndon and P.E. Schupp, {\em Combinatorial group theory}, 
Springer-Verlag, 1977.
 \MR{58:28182}


\bibitem[O1]{O1}
A.Yu. Ol'shanskii, {\em On the Novikov-Adian theorem}, Mat. 
Sbornik {\bf 118} (1982), 203--235.
\MR{83m:20058}


\bibitem[O2]{O2}
A.Yu. Ol'shanskii, {\em Groups of bounded period with subgroups of 
prime order}, Algebra i Logika {\bf 21} (1982), 553--618.
 \MR{85g:20052}


\bibitem[O3]{O3}
A.Yu. Ol'shanskii, {\em Geometry of defining relations in groups}, 
English translation in Math. and Its Applications (Soviet 
series), 70,
Kluwer Acad. Publishers, 1991 (1989).
 \MR{93g:20071}


\bibitem[S]{S}
A.J. Sieradski, {\em Combinatorial isomorphisms and combinatorial 
homotopy equivalences}, J. Pure Appl. Algebra {\bf 7} (1976), 59--65.
 \MR{53:9227}


\bibitem[W]{W}
J.H.C. Whitehead, {\em On adding relations to homotopy groups}, Ann. 
Math. {\bf 42} (1941), 409--428.
\MR{2:323c}

\end{thebibliography}

\end{document}
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