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
*G*_{0}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 |