## 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 $$\sum_i\text{ind}_\Omega(x_i)>2(|\Omega|+g-1)$$.

### 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
Full Text: