Small Profinite Groups

Ludomir Newelski

Profinite groups are inverse limits of finite groups. Small profinite groups are the profinite groups interpretable in small theories.

Question. Is every small profinite group abelian-by-finite ?

Motivated by this question I developed some “model theory” of small profinite groups. The strongest result I proved is as follows.

Theorem. Assume G is a small profinite group, which is m-stable. If G is solvable-by-finite, then G is nilpotent-by-finite.

m-stability is a kind of chain condition. The proof resembles the arguments on groups of finite Morley rank.

An example of a profinite group interpretable in a theory T is the group G/G^{0} for any group G definable in T. If T is superstable, then by the results of Baudisch the group G/G^{0} is solvable-by-finite.

Corollary. If T is a small superstable theory, which is m-stable, then for every group G definable in T, the group G/G^{0} is nilpotent-by-finite.

These results will appear in the paper of mine: Small profinite groups, to appear in J.Symb.Logic. They fit into a more general scheme. By the results of Hrushovski, Pillay and others, any group definable in an ω-stable theory with 1 countable model is abelian-by-finite. I proved the counterpart of this result for profinite groups interpretable in a superstable theory with < 2^{ℵ_{0}} countable models. So one can wonder if the few models assumption may be weakened just to smallness. This is the motivation for the question above.

Address: Wrocław University, pl.Grunwaldzki 2/4, 50-384 Wrocław, Poland

E-mail: newelski@math.uni.wroc.pl