This talk is about the $(a, b, c)$-generation problem for finite simple groups,
where we say that a finite group is an $(a,b,c)$-group if it is a homomorphic image of the
triangle group $$T = T_{a,b,c} = \langle x, y, z : x^a = y^b = z^c = xyz = 1\rangle.$$ Typically, given $T$
(or more generally a Fuchsian group $\Gamma$) and a finite (simple) group $G_0$, one investigates
the following deterministic and probabilistic questions:
- is there an epimorphism in ${\rm Hom}(\Gamma, G_0)$?
- in the case G0 is an (a, b, c)-group, what is the abundance of epimorphisms in ${\rm Hom}(\Gamma,G_0)$?
We first give a short survey of some results in this area where two main methods have been applied: either explicit or probabilistic ones. As a consequence, given a
simple algebraic group $G$ defined over an algebraically closed field of prime characteristic
p, we call $(a,b,c)$ rigid for $G$ if the sum of the dimensions of the subvarieties of $G$
of elements of orders dividing respectively $a$, $b$ and $c$ is equal to $2 \dim G$, and we
conjecture that in this case there are only finitely many integers $r$ such that the finite
group $G_0 = G(p^r)$ of Lie type is a $(a, b,c)$-group. We discuss this conjecture and present a third method we recently developed with Larsen and Lubotzky to study the $(a,b,c)$-generation problem for finite (simple) groups using deformation theory.
This new approach gives some systematic explanation of when finite simple groups of Lie type are quotients of a given $T$.
| When? | 19.03.2013 17:15 |
|---|---|
| Where? | PER 08 Phys 2.52 Chemin du Musée 3 1700 Fribourg |
| Contact | Department of Mathematics isabella.schmutz@unifr.ch |