BEGIN:VCALENDAR VERSION:2.0 PRODID:-//UNIFR/WEBMASTER//NONSGML v1.0//EN CALSCALE:GREGORIAN BEGIN:VEVENT DTSTART;VALUE=DATE:20130319T171500 DTEND;VALUE=DATE:20130319T171500 UID:5666@agenda.unifr.ch DESCRIPTION:This talk is about the $(a, b, c)$-generation problem for finite simple groups,\nwhere we say that a finite group is an $(a,b,c)$-group if it is a homomorphic image of the\ntriangle group $$T = T_{a,b,c} = \langle x, y, z : x^a = y^b = z^c = xyz = 1\rangle.$$ Typically, given $T$\n(or more generally a Fuchsian group $\Gamma$) and a finite (simple) group $G_0$, one investigates\nthe following deterministic and probabilistic questions:\n
    \n
  1. is there an epimorphism in ${\rm Hom}(\Gamma, G_0)$?
    2. \n
  2. in the case G0 is an (a, b, c)-group, what is the abundance of epimorphisms in ${\rm Hom}(\Gamma,G_0)$?
    3. \n
\nWe 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\nsimple algebraic group $G$ defined over an algebraically closed field of prime characteristic\np, we call $(a,b,c)$ rigid for $G$ if the sum of the dimensions of the subvarieties of $G$\nof elements of orders dividing respectively $a$, $b$ and $c$ is equal to $2 \dim G$, and we\nconjecture that in this case there are only finitely many integers $r$ such that the finite\ngroup $G_0 = G(p^r)$ 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.\nThis new approach gives some systematic explanation of when finite simple groups of Lie type are quotients of a given $T$. SUMMARY:Claude Marion (Fribourg): Finite simple quotients of triangle groups CATEGORIES:Colloque / Congrès / Forum LOCATION:PER 08\, Phys 2.52\, Chemin du Musée 3\, 1700 Fribourg URL;VALUE=URI:https://agenda.unifr.ch/e/fr/5666 END:VEVENT END:VCALENDAR