Definably simple groups in o-minimal structures
Definably simple groups in o-minimal structures
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最小结构中的绝对简单群
DOI：
10.1090/s0002-9947-00-02593-9
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发表时间：
2000
影响因子：
1.3
通讯作者：
S. Starchenko
中科院分区：
文献类型：
--
作者：
Y. Peterzil;A. Pillay;S. Starchenko
设G = G，·是可定义在o-极小结构M中的群. G的子集H是G-可定义的，如果H在结构G，·G中可定义（而可定义意味着在结构M中可定义）。设G不存在G可定义的有限指标真子群。本文证明了：若G没有非平凡的阿贝尔正规子群，则G是G-可定义子群H_1，. . .，Hk使得每个Hi可定义地同构于可定义的真实的闭域上的半代数线性群.作为推论，我们得到一个O-最小的类似Cherlin的猜想。这是关于可在o-极小结构中定义的群和真实的闭域上的半代数群的两篇论文中的第一篇。一个O-极小结构是一个结构M = M，<，...其中<是M的稠密线性序，M的任何可定义子集是区间（端点在M <${±∞}）和点的有限并。一个群G被称为在M中可定义的，如果G和G上的群运算的图都是在M中可定义的集合（即M的可定义子集，M对于某个n）。典型的例子是G = H（R），其中H是定义在真实的闭域R上的代数群。(Take M = R，<，+，·，.）我们展示一个匡威：设G可定义在某个o-极小结构中，G是非交换的，且没有可定义在结构G_（？）则G同构于H（R）型群的有限指数（开）半代数子群，其中R是真实的闭域，H是R-单代数群.这给出了（尚未证明的）Cherlin-Zilber猜想的o-极小模拟的肯定答案：任何有限莫利秩的简单群都是代数闭域上的代数群。我们的证明策略与Poizat对Cherlin猜想的方法（[12]）密切相关。给定G在o-极小M中可定义，我们试图找到一个与G密切相连的可在M中定义的真实的闭域R.然后，我们试图证明G是定义（在M）同构于一个线性半代数群R。第一步是可能的，除其他外，三分定理。第二步是发展真实的闭域上的o-极小展开的李理论。这第二步是可能的，因为一旦我们有一个真实的闭域R可定义在一个o-极小结构M中，那么R上的可定义（inM）函数是分段可微的。在实践中，它是方便的工作与无中心和“半单”组，即没有非平凡正常阿贝尔子群的群体，并为这些我们证明收到编辑1998年2月25日。2000年数学学科分类。初级03 C64、22 E15、20 G20;次级12 J15。第二和第三作者部分由NSF支持。2000年美国数学学会American Mathematical Society
Let G = 〈G, ·〉 be a group definable in an o-minimal structure M. A subset H of G is G-definable if H is definable in the structure 〈G, ·〉 (while definable means definable in the structure M). Assume G has no Gdefinable proper subgroup of finite index. In this paper we prove that if G has no nontrivial abelian normal subgroup, then G is the direct product of G-definable subgroups H1, . . . ,Hk such that each Hi is definably isomorphic to a semialgebraic linear group over a definable real closed field. As a corollary we obtain an o-minimal analogue of Cherlin’s conjecture. This is the first of two papers around groups definable in o-minimal structures and semialgebraic groups over real closed fields. An o-minimal structure is a structureM = 〈M,<, ....〉 where < is a dense linear ordering of M , and any definable subset of M is a finite union of intervals (with endpoints in M∪{±∞}) and points. A group G is said to be definable inM if both G and the graph of the group operation on G are definable sets inM (i.e. definable subsets of M, M for some n). The typical example is G = H(R) where H is an algebraic group defined over a real closed field R. (Take M = 〈R, <,+, ·, 〉.) We show a converse: suppose that G is definable in some o-minimal structure and that G is nonabelian and has no proper nontrivial normal subgroup definable in the structure 〈G, ·〉 (we say that G is G-definably simple). Then G is isomorphic to an (open) semialgebraic subgroup of finite index of a group of the form H(R), where R is a real closed field and H is an R-simple algebraic group. This gives a positive answer to the o-minimal analogue of the (yet unproved) Cherlin-Zilber conjecture: any simple group of finite Morley rank is an algebraic group over an algebraically closed field. The strategy of our proof is closely related to Poizat’s approach ([12]) to Cherlin’s conjecture. Given G definable in o-minimalM, we try to find a real closed field R definable inM which is intimately connected to G. We then try to show that G is definably (inM) isomorphic to a linear semialgebraic group over R. The first step is made possible by, among other things, the Trichotomy theorem. The second step goes through developing Lie theory over o-minimal expansions of real closed fields. This second step is possible, because, once we have a real closed field R definable in an o-minimal structureM, then definable (inM) functions on R are piecewise as differentiable as one wants. In practice it is convenient to work with centerless and “semisimple” groups, namely groups with no nontrivial normal abelian subgroups, and for these we prove Received by the editors February 25, 1998. 2000 Mathematics Subject Classification. Primary 03C64, 22E15, 20G20; Secondary 12J15. The second and the third authors were partially supported by NSF. c ©2000 American Mathematical Society