\title{Theta functions and a presentation of $2^{1+(2g+1)}\mathsf{Sp}(2g,2)$}


\author{Dave Benson}
\address{Institute of Mathematics, University of Aberdeen, Aberdeen
  AB24 3UE, Scotland, United Kingdom}

\begin{document}

\begin{abstract}
We investigate a Curtis--Tits style group presentation based on the Dynkin
diagram $C_g$, in which the short root generators have order two while
the long root generators have order four. We prove that this describes a
finite group with an almost extraspecial normal subgroup of order
$2^{2g+2}$ and quotient isomorphic to the symplectic group 
$\Sp(2g,2)$. Such an
extension has to be non-split, and was abstractly proved to exist in a paper of
Griess from 1973. Our presentation proves that it is a double
cover of a finite quotient of $\Sp(2g,\bZ)$. We investigate a $2^g$
dimensional complex representation on a suitable space of theta
functions, and produce some consequences for the signatures of
$4$-manifolds described as surface
bundles over surfaces. In particular, we prove that if the monodromy
is contained in the theta subgroup $\Spq(2g,\bZ)\leq\Sp(2g,\bZ)$ then 
the signature of the $4$-manifold is divisible by eight.
\end{abstract}