Moscow-Petersburg Seminar on Low-Dimensional Mathematics
September 17, 2004

M.Vsemirnov (SPb-Cambridge)

Hurwitz groups of low rank.

Finite homomorphic images of the triangle group
  $$ T(2,3,7)= \langle X, Y : X^2=Y^3=(XY)^7=1 \rangle $$
are called Hurwitz groups. They arise, for example, in the theory of Riemann
surfaces  as the groups of automorphisms of the maximal size with respect to
the genus. It is known (Lucchini, Tamburini, Wilson) that the most of the series
of classical finite groups are Hurwitz provided their rank is large enough.
For instance, $SL(n,q)$ are Hurwitz for any $n>286$ and any $q$. For small
ranks the situation is more complicated. Di Martino, Tamburini and
Zalesski found natural constraints which are consequences of Scott's
formula.

In the talk some new results due to the speaker and M. C. Tamburini
will be discussed. The following ones are among them:

1) 60 new values of $n$ such that $SL(n,q)$ are Hurwitz.

2) Explicit Hurwitz generators for $G_2(3^m)$, $m>1$ and orthogonal
  groups $\Omega_{8}^{\pm}(q)$, $q=p^m$, $p>3$.

3) Parameterization of Hurwitz pairs in $SL(n,F)$ and $PSL(n,F)$, $n<8$.

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