Seminar on low-dimensional mathematics
March 3, 2006

Maxim Vsemirnov. 
Is the group SL(6,Z) (2,3)-generated?

A group is called (2,3)-generated if it can be generated by an involution
and an element of order 3. These groups can be also characterized as
homomorphic images of PSL(2,Z). The class of (2,3)-generated groups
is extremely large and any classification is hopeless. Thus, some
restricted classes of (2,3)-generated groups are studied.

For the groups SL(n,Z) the answer, which is close to be optimal, was
given by Tamburini and Sacchini. Namely, they proved that SL(n,Z) is
(2,3) generated for any n>12. On the other hand, SL(2,Z), SL(3,Z),
SL(4,Z) are not (2,3)-generated (this is rather trivial for n=2,4,
and non-trivial for n=3). For 4<n<13 the problem remains open.
Luzgarev and Pevzner had a partial progress in the case n=5.

In the talk we address the case n=6. We show that the problem can
be reduced to the question whether SL(6,Z) is generated by one
of the eight explicit pairs of matrices. If time permits we also
show how two cases can be eliminated further and we discuss
some difficulties arising in the six remaining cases.

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