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**Composition factors of monodromy groups.**
*(English)*
Zbl 1004.20001

In 1990, R. M. Guralnick and J. G. Thompson formulated the following conjecture: For each nonnegative integer \(g\) there is a finite set \(E(g)\) of finite simple groups such that if \(X\) is a compact Riemann surface of genus \(g\), \(\Phi\colon X\to P^1\mathbb{C}\) is a cover, and \(S\) is a nonabelian composition factor of the monodromy group \(\text{Mon}(X,\Phi)\), then \(S\) is either an alternating group or isomorphic to a member of \(E(g)\).

Using Riemann’s existence theorem, the conjecture can be translated into a problem on primitive permutation groups, which in turn has been reduced to a question about actions of classical groups on subspaces of their natural modules. Using some of their earlier results concerning estimates of fixed point ratios together with a result of L. Scott on the generation of linear groups the authors prove the following permutation group formulation of the Guralnick-Thompson conjecture.

Theorem A. For each nonnegative integer \(g\) there is a finite set \(E(g)\) of finite simple groups such that if \(G\) is a transitive group of permutations on the finite set \(\Omega\) satisfying (A.1) \(G=\langle x_1,\dots,x_r\rangle\) for some \(x_i\in G\), (A.2) \(x_1x_2\cdots x_r=1\), (A.3) \(\sum_i\text{ind}_\Omega(x_i)=2(|\Omega|+g-1)\) (here \(\text{ind}_\Omega(x_i):=|\Omega|-\text{orb}_\Omega(x_i)\) with \(\text{orb}_\Omega(x_i)\) denoting the number of orbits of \(x_i\) on \(\Omega\)), and \(S\) is a composition factor of \(G\) such that \(S\) is neither alternating nor cyclic then \(S\in E(G)\).

The proof of Theorem A is based on the following Theorem B. Let \(G\) be an almost simple finite classical group with natural module \(V\), and let \(g\) be a nonnegative integer. Assume that \(G\) acts primitively and faithfully on a set \(\Omega\) of subspaces of \(V\). Assume further that \(x_1,\dots,x_r\) are generators of \(G\) such that \(x_1x_2\cdots x_r=1\). If \(\dim(V)\) is sufficiently large, then \(\sum_i\text{ind}_\Omega(x_i)>2(|\Omega|+g-1)\).

Using Riemann’s existence theorem, the conjecture can be translated into a problem on primitive permutation groups, which in turn has been reduced to a question about actions of classical groups on subspaces of their natural modules. Using some of their earlier results concerning estimates of fixed point ratios together with a result of L. Scott on the generation of linear groups the authors prove the following permutation group formulation of the Guralnick-Thompson conjecture.

Theorem A. For each nonnegative integer \(g\) there is a finite set \(E(g)\) of finite simple groups such that if \(G\) is a transitive group of permutations on the finite set \(\Omega\) satisfying (A.1) \(G=\langle x_1,\dots,x_r\rangle\) for some \(x_i\in G\), (A.2) \(x_1x_2\cdots x_r=1\), (A.3) \(\sum_i\text{ind}_\Omega(x_i)=2(|\Omega|+g-1)\) (here \(\text{ind}_\Omega(x_i):=|\Omega|-\text{orb}_\Omega(x_i)\) with \(\text{orb}_\Omega(x_i)\) denoting the number of orbits of \(x_i\) on \(\Omega\)), and \(S\) is a composition factor of \(G\) such that \(S\) is neither alternating nor cyclic then \(S\in E(G)\).

The proof of Theorem A is based on the following Theorem B. Let \(G\) be an almost simple finite classical group with natural module \(V\), and let \(g\) be a nonnegative integer. Assume that \(G\) acts primitively and faithfully on a set \(\Omega\) of subspaces of \(V\). Assume further that \(x_1,\dots,x_r\) are generators of \(G\) such that \(x_1x_2\cdots x_r=1\). If \(\dim(V)\) is sufficiently large, then \(\sum_i\text{ind}_\Omega(x_i)>2(|\Omega|+g-1)\).

Reviewer: Wolfgang Lempken (Essen)

### MSC:

20B15 | Primitive groups |

14H30 | Coverings of curves, fundamental group |

20D05 | Finite simple groups and their classification |

30F10 | Compact Riemann surfaces and uniformization |

20G40 | Linear algebraic groups over finite fields |