


%  Put the relevent information between the braces below:

\titleofpresentation{The geometry of Zappa-Szep products of groups}

\presenter{Stephen J Pride}  %  Name just one speaker, please; collaborators
            %  can be credited in the body of the abstract.

A group $G$ is a \emph{semidirect product} of its two subgroups
$H$ and $K$ if: 
\begin{enumerate}
\item
$H$ is normal in $G$;
\item
$H \cap{K} = 1$;
\item
$HK=G$. 
\end{enumerate}
In such a situation we have a homomorphism
\begin{equation*}
\psi : K \longrightarrow Aut(H),\ \psi_ {k} (h)=khk^{-1} \ (h\in
H, k\in K),
\end{equation*}
and then the multiplication is given by
\begin{equation*}
(hk)(h'k')=h\psi_{k}(h') kk'.
\end{equation*}
Conversely, given a pair of groups $H, K$ and a homomorphism $\psi
: K \longrightarrow Aut(H)$, we can construct a group $\hat{G}$
consisting of the ordered pairs $(h, k)$ with multiplication
\begin{equation*}
(h,k)(h',k')=(h\psi_{k}(h'), kk').
\end{equation*}
Then $\hat{H} = \{(h,1): h\in H\},\ \hat{K} = \{(1,k): k\in K\}$
are isomorphic copies of $H, K$, and $\hat{G}$ is the semidirect
product of $\hat{H}, \hat{K}$.

A \emph{Zappa-Szep product} occurs when condition (1) above is not
required to hold. To \emph{construct} such a product from given
groups $H, K$, two functions $K\times H \longrightarrow K$,
$K\times H \longrightarrow H$ are needed with certain conditions.
Most of the work on this construction has been algebraic (see
\cite{pride1,pride2} and the references cited there). However, using
techniques of
geometric group theory one can get a different perspective on the
construction. I will describe this in my talk, and use such
techniques to obtain new results concerning these groups.

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

\bibitem{pride1} Matthew G. Brin, \emph{On the Zappa-Szep
product}, Comm. in Alg. \textbf{33} (2005), 393-424.

\bibitem{pride2} M. Kunze, \emph{Zappa
products}, Acta Math. Hung. \textbf{41} (1983), 225-239.

\bibitem{pride3} Stephen J Pride, \emph{The geometry of Zappa-Szep
products}, in preparation.

\end{ourbiblio}

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

\email{sjp@maths.gla.ac.uk}



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