Group of finite Morley rank

A group $G$ such that the formula $x=x$ has finite Morley rank in the theory $\mathrm{Th}(G,{\cdot})$. (Cf. also Model theory.) Sometimes $G$ appears as a definable group in a structure $M$ and in this context $G$ is said to have finite Morley rank if the formula defining $G$ in $M$ has finite Morley rank with respect to the theory of $M$.

There is a well-developed theory of groups of finite Morley rank, both from the model-theoretic and group-theoretic point of view. The theory began with B.I. Zil'ber's study [a4] of groups definable in uncountably categorical structures. G.L. Cherlin's paper [a2] also played an important role in the early theory. Examples of groups of finite Morley rank are algebraic groups over algebraically closed fields. The Cherlin–Zil'ber conjecture says that any infinite non-commutative simple group of finite Morley rank is an algebraic group over an algebraically closed field. The conjecture remains unproved (1996). A certain amount of the theory of algebraic groups can be developed for groups of finite Morley rank, specifically the notions of generic type, connected component, and stabilizer. Another important technical tool is the Zil'ber indecomposability theorem, which states that if $G$ is a group of finite Morley rank and $X_i$, for $i\in I$, is a family of definable subsets of $G$ satisfying some mild assumptions, then the subgroup of $G$ generated by all the $X_i$ is definable and connected. This is an analogue of the Borel theorem for algebraic groups.

The relevance of groups of finite Morley rank for model theory comes from a theorem of Zil'ber which states that if $M$ is a model of an uncountably categorical theory (cf. Categoricity in cardinality), then $M$ is built up from a set of Morley rank $1$ by a finite sequence of "definable fibre bundles" .

Much of the recent work on the Cherlin–Zil'ber conjecture is contained in [a1].

A vast generalization of the theory of groups of finite Morley rank is the theory of stable groups, due essentially to B. Poizat [a3].

References

[a1]	A. Borovik, A. Nesin, "Groups of finite Morley rank" , Oxford Univ. Press (1994) MR1321141 MR1273276 MR1273275 Zbl 0816.20001 Zbl 0818.20031 Zbl 0818.20030
[a2]	G. Cherlin, "Groups of small Morley rank" Ann. Math. Logic , 17 (1979) pp. 1–28 MR0552414 Zbl 0427.20001
[a3]	B. Poizat, "Groupes stables" , Nur Al-Mantiq Wal-Ma'rifah, Villeurbanne (1987) MR0902156 MR0895648 Zbl 0633.03019 Zbl 0626.03025
[a4]	B.I. Zil'ber, "Groups and rings whose theory is categorical" Fundam. Math. , 55 (1977) pp. 1730188 Zbl 0745.03029