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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<ol style="list-style-type: lower-alpha">\n<li>is there an epimorphism in <span class="math">${\rm Hom}(\Gamma, G_0)$</span>?</li>\n<li>in the case <span class="math"><em>G</em><sub>0</sub></span> is an <span class="math">(<em>a</em>, <em>b</em>, <em>c</em>)</span>-group, what is the abundance of epimorphisms in <span class="math">${\rm Hom}(\Gamma,G_0)$</span>?</li>\n</ol>\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
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