Presentations, Cohomology and Representations of Finite Groups and Coverings of Curves

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The PI plans to study some basic problems about finite and algebraic groups related to presentations, linear and permutation representations, and cohomology with applications to the problems in arithmetic algebraic geometry--particularly questions related to polynomials, rational functions, and automorphisms and coverings of curves. Using the work of Tiep and the PI, small representations of simple groups will be studied. This work has already had applications to various questions of Katz, Kollar, and Larsen in algebraic geometry. The most critical open case is to classify all closed subgroups which are irreducible on exterior powers of a module. This is also closely related to questions about the classification of maximal subgroups of the finite simple groups. This latter questions leads to many basic problems. In particular the PI will look at problems about bounding the number of irreducible representations of bounded dimension for various families of simple groups.The PI and his collaborators have shown recently that with the possible exception of one family, every finite simple group has a presentation with at most fifty relations. In many cases, one can do much better (for example, an infinite number of alternating groups can be presented with two generators and four relations). Allowing the number of relations to increase a bit, one can even produce short presentations (essentially best possible). The most fundamental problem is the relationship between discrete presentations and profinite presentations and cohomology (one way to think of a profinite presentation is that if one is given by generators and relations and one already knows the group is finite, then one can identify the group). This leads one to try to produce very good bounds on the size of the first and second cohomology groups of finite and algebraic groups with coefficients in a simple module. One goal is to prove that every finite simple group has a profinite presentation with two generators and at most four relations (one cannot do better). This may be even be true without the profiniteness condition but no one has any idea of how to approach this in general. Another major problem is to complete the classification of exceptional polynomials over finite fields. Exceptional polynomials are precisely the bijective polynomials assuming the field size is sufficiently large compared to the degree and have been studied seriously since the thesis of Dickson in the 1890's, as well as by Schur, Fried and others. The classification of indecomposable exceptional polynomials whose degree is not a power of the characteristic used a combination of deep group theory together with methods arithmetic algebraic geometry. The PI intends to use these methods and use new results to study these problems. This should have applications to cryptography.