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We describe a new algorithm for computing braid orbits on Nielsen classes. As an application we classify all families of affine genus zero systems; that is all families of coverings of the Riemann sphere by itself such that the monodromy group is a primitive affine permutation group.
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We prove that for a connected, semisimple linear Lie group $G$ the spaces of generating pairs of elements or subgroups are well-behaved in a number of ways: the set of pairs of elements generating a dense subgroup is Zariski-open in the compact case, Euclidean-open in general, and always dense. Similarly, for sufficiently generic circle subgroups $H_i$, $i=1,2$ of $G$, the space of conjugates of $H_i$ that generate a dense subgroup is always Zariski-open and dense. Similar statements hold for pairs of Lie subalgebras of the Lie algebra $Lie(G)$.
Denote by $m(G)$ the largest size of a minimal generating set of a finite group $G$. We estimate $m(G)$ in terms of $sum_{pin pi(G)}d_p(G),$ where we are denoting by $d_p(G)$ the minimal number of generators of a Sylow $p$-subgroup of $G$ and by $pi(G)$ the set of prime numbers dividing the order of $G$.
For a group $G,$ let $Gamma(G)$ denote the graph defined on the elements of $G$ in such a way that two distinct vertices are connected by an edge if and only if they generate $G$. Moreover let $Gamma^*(G)$ be the subgraph of $Gamma(G)$ that is induced by all the vertices of $Gamma(G)$ that are not isolated. We prove that if $G$ is a 2-generated non-cyclic abelian group then $Gamma^*(G)$ is connected. Moreover $mathrm{diam}(Gamma^*(G))=2$ if the torsion subgroup of $G$ is non-trivial and $mathrm{diam}(Gamma^*(G))=infty$ otherwise. If $F$ is the free group of rank 2, then $Gamma^*(F)$ is connected and we deduce from $mathrm{diam}(Gamma^*(mathbb{Z}times mathbb{Z}))=infty$ that $mathrm{diam}(Gamma^*(F))=infty.$
We show that if $cal S$ is a compact Riemann surface of genus $g = p+1$, where $p$ is prime, with a group of automorphisms $G$ such that $|G|geqlambda(g-1)$ for some real number $lambda>6$, then for all sufficiently large $p$ (depending on $lambda$), $cal S$ and $G$ lie in one of six infinite sequences of examples. In particular, if $lambda=8$ then this holds for all $pgeq 17$ and we obtain the largest groups of automorphisms of Riemann surfaces of genenera $g=p+1$.
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