Triangle generation of finite exceptional groups of low rank.
Given a triple \((p_1,p_2,p_3)\) of primes, let \(T\) be the triangle group with presentation \(T=\langle x,y,z:\;x^{p_1}=y^{p_2}=z^{p_3}=xyz=1\rangle\). The author is concerned with determining the abundance of epimorphisms in the space \(\Hom(T,G)\) of homomorphisms from \(T\) to a finite simple exceptional group of Lie type of low rank. If \(1/p_1+1/p_2+1/p_3\geq 1\), then the only non-Abelian finite simple image of \(T\) is the alternating group \(A_5\). So one can restrict the attention to the case when \(1/p_1+1/p_2+1/p_3<1\), so that \(T\) is a hyperbolic triangle group. With a few exceptions, the author gives precise asymptotic estimates for the size of \(\Hom(T,G)\) and determines the limiting probability that a randomly chosen homomorphism from \(T\) to \(G\) is surjective as \(|G|\to\infty\). The proofs use character theory of \(G\) to determine a precise asymptotic estimate for \(|\Hom(T,G)|\) and an exam of the maximal subgroups of \(G\) to get upper bounds for \(|\bigcup_M\Hom(T,M)|\) where \(M\) varies through all maximal subgroups of \(G\). This enables the author to estimate \(1-|\Hom(T,G)|^{-1}|\bigcup_M\Hom(T,M)|\), which is by definition the probability that a randomly chosen element in \(\Hom(T,G)\) is surjective.
scientific article; zbMATH DE number 5969517
scientific article