

%  Put the relevent information between the braces below:

\titleofpresentation{On torsion in free central extensions of groups}

\presenter{Ralph~St\"{o}hr}  %  Name just one speaker, please; collaborators
            %  can be credited in the body of the abstract.

Let $G$ be a group given by a free presentation $G=F/R$ and consider
the quotient
\begin{equation}\label{stohr1}
    F/[\gamma_cR,F]
\end{equation}
where $\gamma_cR$ denotes the $c$-th term of the lower central
series of $R$ and $c\geq 2$. This quotient is a free central
extension of $F/\gamma_cR$, which is in its turn an extension of
$G=F/R$ with free nilpotent kernel $R/\gamma_cR$. While the quotient
$F/\gamma_cR$ is always torsion-free, elements of finite order may
occur in $\gamma_cR/[\gamma_cR,F]$, the centre of \eqref{stohr1}. In the
case where $c=2$ and $R=F'$, that is for the free
centre-by-metabelian group $F/[F'',F]$, this was   discovered by
C.K. Gupta who proved in 1973 that the free centre-by metabelian
group of rank $n\geq 4$ contains an elementary abelian $2$-group of
rank $\binom{n}{4}$ in its centre. This was a major surprise at the
time as it was the first example of a relatively free group that is
given by a single multilinear commutator identity and is not
torsion-free. In a pioneering paper of 1977 Yu.V. Kuz'min identified
Gupta's torsion subgroup with the fourth homology group of the free
abelian group $F/F'$ reduced modulo $2$, thus linking the problem to
homology of groups. In the eighties there was a series of papers by
Yu.V. Kuz'min focusing on the case $c=2$, and there was a series of
papers by the speaker, including a joint paper \cite{Stohr1} with
the late Brian Hartley, on the general case $c\geq 2$. One of the
main achievements of all these papers was a precise identification
of the torsion subgroup $\text{t}(\gamma_cR/[\gamma_cR,F])$ of
\eqref{stohr1} in the case when $c$ is a prime, say $c=p$, in terms of
homology.  Namely, if $G$ has no elements of order $p$, then
\begin{equation*}
        \text{t}(\gamma_pR/[\gamma_pR,F])\cong H_4(G,\mathbb{Z}_p)
\end{equation*}
where $\mathbb{Z}_p=\mathbb{Z}/p\mathbb{Z}$ regarded as a trivial
$G$-module. In the early nineties there was a single addition to
this body of results for the case $c=4$: for $G$ without elements of
order $2$ there is an isomorphism
$\text{t}(\gamma_4R/[\gamma_4R,F])\cong H_6(G,\mathbb{Z}_2)$. This
was the last word on the subject for a long time. The methods used
in the eighties and nineties appeared to be exhausted. Very
recently, new results by Bryant, Erdmann and Schocker on modular Lie
representations made it possible to make further progress on the
torsion subgroup of \eqref{stohr1}. In joint work with Marianne Johnson
we have been able to identify this torsion subgroup for $c=6$ (for
$G$ without elements of order $2$ and $3$). The result was a big
surprise to us. In my talk I will review the history of the problem,
and then I will focus on these recent developments.

\begin{ourbiblio}{99}  %  Delete this environment if you
               %  aren't giving a bibliography.

\bibitem{Stohr1} Hartley, B. ;  St\"{o}hr, R.  Homology of higher relation modules
and torsion in free central
 extensions of groups.
 Proc. London Math. Soc. (3)  62  (1991),  no. 2, 325--352.



\end{ourbiblio}



\presenteraddress{University of Manchester}  % The name of a university is fine.

\email{ralph.stohr@manchester.ac.uk}

\web{www.maths.manchester.ac.uk/~rs/}  % Those without homepages should delete this.

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