Euler characteristic in semialgebraic and other o-minimal groups

Abstract

Semialgebraic sets (subsets of R n defined by polynomial inequalities) and (discontinuous) semialgebraic maps form a category with many of the desirable properties of the category of finite sets, suggesting that groups in this category should be somewhat like finite groups. We develop this idea, in the more general setting of the category of definable sets and maps in an o-minimal structure. In this category Euler characteristic plays the role played by cardinality in the category of finite sets. We generalize Sylow's theorems to definable groups, with some new features arising: there are 0-groups as well as p-groups. Introducing the notion of parametrizable sets of definable subgroups allows us to generalize the Sylow-Frobenius theorem. We prove several properties of definable groups; among them are: groups with bounded exponent are finite (Proposition 6.1), groups of Euler characteristic one are uniquely divisible (Proposition 4.1), definable 0-groups are abelian (Corollary 5.17), semialgebraic 0-groups are monogenic, and have only countably many “maximal” subgroups (Corollaries 5.14 and 5.15), maximal tori are conjugate (Corollary 5.19). We also give some examples of “exotic” groups.