14 research outputs found

    Short Laws for Finite Groups of Lie Type

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    We produce new short laws in two variables valid in finite groups of Lie type. Our result improves upon results of Kozma and the second named author, and is sharp up to logarithmic factors, for all families except possibly the Suzuki groups. We also produce short laws valid for generating pairs and random pairs in finite groups of Lie type, and, conditional on Babai's diameter conjecture, make effective the dependence of our bounds on the rank. Our proof uses, among other tools, the Classification of Finite Simple Groups, Aschbacher's structure theorem for maximal subgroups for classical groups, and upper bounds on the diameters of finite simple groups due to Breuillard, Green, Guralnick, Pyber, Szabo and Tao.Comment: 38 pages, comments welcom

    Subgroups of cremona groups

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    The Cremona group in n-variables Cr_n(C) is the group of birational transformations of the complex projective n-space. This thesis contributes to the research on Cremona groups through the study of certain classes of „large'' subgroups. In the first part we consider algebraic embeddings of Cr_2(C) into Cr_n(C). In particular, we describe geometrical properties of an embedding of Cr_2(C) into Cr_5(C) that was discovered by Gizatullin. We also classify all algebraic embeddings from Cr_2(C) into Cr_3(C), and we partially generalize this result to embeddings of Cr_n(C) into Cr_{n+1}(C). In a second part, we look at degree sequences of birational transformations of varieties over arbitrary fields. We show that there exist only countably many such sequences and we give new obstructions on the degree growth of automorphisms of affine n-space. In the third part, we classify subgroups of Cr_2(C) containing only elliptic elements, i.e. elements whose iterates are of bounded degree. From this we deduce in particular the Tits alternative for arbitrary subgroups of Cr_2(C). In the last part, we show that every finitely generated simple subgroup of Cr_2(C) is finite and, under the hypothesis of an unproven conjectural lemma, that a simple group can be embedded into Cr_2(C) if and only if it can be embedded into PGL_3(C)
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