SUPERSIMPLE ω-CATEGORICAL GROUPS

FRANK O WAGNER

Abstract. An ω-categorical supersimple group is finite-by-abelian-by-finite, and has finite SU-rank. Every definable subgroup is commensurable with an acl(∅)- definable subgroup.

It is well-known that an ω-categorical superstable group is abelian-by-finite [1] and has finite rank (in fact by [3] every ω-categorical superstable theory is even one-based of finite Morley rank). In the case of a supersimple ω-categorical theory, much less is known except that things do not turn out that nicely: recently, Hrushovski [5] has constructed a simple ω-categorical theory of SU-rank 1 whose geometry is not locally modular. It is unknown whether a supersimple ω-categorical theory can have infinite rank. The question therefore arises whether at least in the group case the results from the superstable case generalize. This is indeed the case; we shall prove the following:

Theorem 1. An ω-categorical supersimple group is finite-by-abelian-by-finite, and has finite SU-rank.

The proof will proceed in two stages: we shall first prove that an ω-categorical group of finite SU-rank is finite-by-abelian-by-finite. (That the argument given in [7] for ω-categorical groups of finite Morley rank can be generalized to the present context has also been noted by David Evans [4].) For this we shall also need some information about definable subgroups of an ω-categorical group of finite SU-rank. It turns out that as in the superstable case, every definable subgroup is commensurable with one definable over acl(∅). However, it is not true that an ω-categorical group of finite SU-rank is necessarily one-based: Pourmahdian [8] has fused Hrushovski’s example mentioned above with an elementary abelian 2-group, and obtained an ωcategorical group of SU-rank 1 which is not one-based. Finally, we shall use the information obtained for ω-categorical groups of finite SU-rank to show that ωcategorical supersimple groups of infinite rank cannot exist.

One of our main tools is the following Theorem due to Schlichting [9], and rediscovered by Bergmann and Lenstra [2]:

Fact 2. Let G be a group and 𝔉 a family of subgroups such that for any F_{1}, F_{2} ∈ 𝔉 the index |F_{1} : F_{1} ∩ F_{2}| is bounded. Then there is a subgroup F^{c}, invariant under all automorphisms of G which stabilize 𝔉 setwise, and commensurable with all F ∈ 𝔉.

Research at MSRI is supported in part by NSF grant DMS-9701755.

Heisenberg-Stipendiat der Deutschen Forschungsgemeinschaft (Wa 899/2-1).

1991 Mathematics Subject Classification: 03C45.

In fact, it is easy to see that if all the elements of 𝔉 are definable, then F^{c} is definable as a finite extension of a finite intersection of groups in 𝔉. (In our case definability also follows immediately from ω-categoricity.)

Corollary 3. Let F be a definable subgroup of an ω-categorical group. Then there is a definable subgroup F^{c}, commensurable with F, such that F^{c} is stabilized under any automorphism of G stabilizing the commensurability class of F.

Proof: By ω-categoricity, there is a bound on the indices |F : F ∩ F′|, for F′ a commensurable conjugate of F. Now apply Fact 2. □

In fact, the same conclusion holds for any relatively definable subgroup of a group type-definable in a simple theory, see [10, Corollary 1.8]. Clearly, F^{c} is defined over the same parameters as the original F.

We shall first look at abelian groups of finite SU-rank. The following definition differs from the stable case.

Definition 1. A group is minimal if every definable subgroup of infinite index is finite.

Note that in a supersimple theory every definable group contains a definable minimal subgroup (any infinite definable subgroup of minimal possible SU-rank will do).

Theorem 4. Let G be an ω-categorical finite-by-abelian-by-finite group of finite SUrank. Then any definable subgroup of G is commensurable with one definable over acl(∅).

Proof: Suppose N is a normal subgroup of finite index in G and with finite commutator group N′. If N_{1} is a conjugate of N by some automorphism of G, then (NN_{1})′ is finite as well; we may thus assume that N is characteristic, and hence ∅-definable, as is N′. Replacing a definable subgroup H of G by the definable subgroup (H ∩ N)/N′ of N/N′, we can reduce the problem to the case when G is abelian.

For abelian G, consider first a minimal subgroup A of G. By the finiteness of rank, there are finitely many conjugates of A, say (A_{i} : i < n), such that every conjugate of A intersects A^{0} := ∑_{i<n} A_{i} in a subgroup of finite index. We may choose the A_{i} almost linearly independent, i.e. A_{i} ∩ ∑_{j≠i} A_{j} is finite for all i < n.

Definition 2. A virtual isogeny f between an abelian minimal group A and an abelian minimal group B is a definable isomorphism f : D/K → I/C, where the domain D has finite index in A, the image I has finite index in B, and K (the kernel) and C (the cokernel) are finite. Two virtual isogenies f_{1} and f_{2} are equivalent if the derived maps from D_{1} ∩ D_{2} to (I_{1} + I_{2})/(C_{1} + C_{2}) agree on a subgroup of finite index in A. This is an equivalence relation, and in fact a congruence.

Lemma 5. In an ω-categorical theory, the equivalence class of every virtual isogeny from A to B (both abelian minimal ∅-definable groups) is algebraic over ∅ (as an imaginary element).

Proof: It is standard that the family of virtual isogenies from A to A modulo equivalence, together with the zero endogeny, forms a ring R with identity, called the ring of virtual endogenies; it is easy to see from minimality that R has no zero divisors. If A is ω-categorical, R must be locally finite, whence a locally finite field, and every element of R is algebraic. Furthermore, if there is a virtual isogeny f between A and B, then composing any virtual isogeny r between A and B with f^{−1} (which is a virtual isogeny between B and A) yields a virtual isogeny of A; clearly r is algebraic over rf^{−1} and f. Hence the equivalence class [r] of r is algebraic over [f]. Choosing another virtual isogeny f′ between A and B with acl(f′) ∩ acl(f) = acl(∅), we see that [r] must be algebraic over ∅. □

Now consider any conjugate A′ of A. Every a ∈ A′ ∩ A^{0} can be written as ∑_{i<n} a_{i} with a_{i} ∈ A_{i}; since A′ ∩ A^{0} is infinite by maximality of n, there must be some i = i(A′) < n such that A_{i} ∩ (A′ + ∑_{k≠i} A_{k}) is infinite. For every j ≠ i we now define an endogeny r(A′, j) from A_{i} to A_{j}: if A_{j} ∩(A′ +∑_{k≠j} A_{k}) is finite, then r(A′, j) = 0; otherwise r(A′, j)(x) := {y ∈ A_{j} : x−y ∈ A′ +∑_{k≠i,j} A_{k}} = A_{j} ∩(x+A′ +∑_{k≠i,j} A_{k}) (and it is easy to see that this is a virtual isogeny). But if A′′ is such that i(A′′) = i(A′), and r(A′, j) and r(A′′, j) are equivalent virtual endogenies for all j ≠ i(A′), then one can check that A′ and A′′ must be commensurable. It follows that there are only finitely many commensurability classes among the conjugates of A.

Consider the group A^{c} given by Corollary 3. It has only finitely many conjugates, corresponding to the commensurability classes of conjugates of A, and must therefore be definable over acl(∅). This proves the assertion for minimal groups.

If H ≤ G is not minimal, then by supersimplicity it contains a minimal subgroup A which is commensurable with some acl(∅)-definable A^{c}. But HA^{c}/A^{c} is a subgroup of G/A^{c} of smaller SU-rank; by induction it is commensurable with an acl(∅)-definable group H_{c}/A^{c}, whose preimage H^{c} in G is as required. □

Theorem 6. An ω-categorical group of finite SU-rank is finite-by-abelian-by-finite.

Proof: Suppose not, and let G be a counterexample. By ω-categoricity, we may assume that G has no proper ∅-definable subgroups of finite index.

Claim: We may assume that G′/(G′ ∩ Z(G)) is finite.

Proof of Claim: By simplicity, G does not have the strict order property; by [6, Theorem 1.2] it is nilpotent. By ω-categoricity, the subgroups G^{n} in the lower central series are ∅-definable and there is some minimal n > 1 such that G′/G^{n} is infinite. (Note that G/G′ must be infinite since G has no characteristic subgroup of finite index, and G′ is infinite since G is not finite-by-abelian.)

We claim that G/G^{n} is not finite-by-abelian-by-finite. So suppose otherwise, and let N be a normal subgroup of G of finite index (not necessarily definable) such that N′/G^{n} is finite. If N_{1} is a conjugate of N by some automorphism of G, then (NN_{1})′/G^{n} is finite as well; we may thus assume that N is characteristic. But since G has no proper characteristic subgroup of finite index, N = G, and G′/G^{n} is finite, a contradiction.

As G′/G^{n−1} is finite by minimality of n, G^{n−1}/G^{n} is infinite; since G^{n−1}/G^{n} is contained in Z(G/G^{n}), we may replace G by G/G^{n} and are done. □

Since G′/(G′ ∩ Z(G)) is finite, G_{g} := C_{G/Z(G)}(g/Z(G)) has finite index in G/Z(G) for every g ∈ G. Put H_{g} := {(h, [h, g]) : h ∈ G_{g}}, a subgroup of G/Z(G) × Z(G)

(note that the commutator [h, g] is well-defined). Since G/Z(G) × Z(G) is finite-by-abelian, H_{g} is commensurable with an acl(∅)-definable group by Lemma 4; it follows by compactness that there are only finitely many commensurability classes in the family {H_{g} : g ∈ G}. Note that [h, g] = [h, g′] for any h ∈ π_{1}(H_{g} ∩ H_{g′}), whence [h, g′g^{−1}] = 1. However, we may choose g and g′ to be two independent generic elements such that H_{g} and H_{g′} are commensurable. Then π_{1}(H_{g} ∩ H_{g′}) is a subgroup of finite index in G/Z(G), and g^{−1}g′ is a generic element of G with |G : C_{G}(g^{−1}g′)| finite.

The set of all g ∈ G such that C_{G}(g) has finite index in G is a subgroup of G, which is ∅-definable by ω-categoricity; by the above it has finite index in G. As G has no ∅-definable subgroups of finite index by assumption, C_{G}(g) has finite index in G for all g ∈ G. But then [g, G] is finite for all g ∈ G. If g | ⌣ g′, then [g, g′] ∈ acl(g) ∩ acl(g′) = acl(∅). Since every element g′ of G can be written as g′ = g_{1}g_{2} with g | ⌣ g_{1} and g | ⌣ g_{2}, we obtain [g, g′] = [g, g_{1}g_{2}] = [g, g_{2}][g, g_{1}][g, g_{1}, g_{2}]. Then [g, g_{1}] ∈ acl(∅) is independent of g_{2}, and [g, g_{1}, g_{2}] ∈ acl(∅) as well. Thus there are only finitely many commutators and G′ is finite, a contradiction. □

Theorem 7. An ω-categorical supersimple group has finite SU-rank.

Proof: Let G be an ω-categorical supersimple group of infinite rank, and let φ(x) be a formula isolating a type p of rank ω. Let n be a maximal finite rank of a 1-type over the parameters used so far.

Let ā be a tuple of independent realizations of p such that there is a non-algebraic type of finite SU-rank greater than n over ā. (For example, take a forking extension q of p of rank greater than n; a sufficiently long initial segment of a Morley sequence in q will do.) Let p̄ be the type of ā, and let m be the maximal rank of an extension of p of finite SU-rank over 2·|ā| elements. Let q̄ be a non-algebraic extension of p̄ of finite SU-rank, such that the SU-rank of every co-ordinate over the domain of q̄ and the other co-ordinates is greater than m. Consider a Morley sequence I := (ā_{i} : i < ω) in q̄. Then ā_{0} | ⌣ ā_{1} by assumption on the SU-ranks of the co-ordinates of q̄. Choose a maximal independent subset A of I containing ā_{0} (independent as singletons); this is a finite set. Let H := {g ∈ G : SU(g/ā_{0}) < ω} and K := {g ∈ G : SU(g/A) < ω}. By ω-categoricity, H is an ā_{0}-definable subgroup and K is an A-definable subgroup of G, with H ≤ K and SU(K) < ω.

As I is a Morley sequence, for every j < ω there is an automorphism σ_{j} of G which maps ā_{i} to ā_{i+j} for all i < ω. Then SU(σ_{j}(A)) = SU(A) = ω · |A|, but SU(ā_{0}, …, ā_{n}) < SU(A) + ω for all n < ω. It follows that SU(A/σ_{j}(A)) < ω, so σ_{j} stabilizes K (setwise). However, it maps ā_{0} to ā_{j}. By Theorem 4 there are only finitely many commensurability classes among K-conjugates of H, so there is i < j such that σ_{i}(H) and σ_{j}(H) are commensurable, whence σ_{i}(H^{c}) = σ_{j}(H^{c}).

Now H^{c} is definable over ā, so σ_{i}(H^{c}) is definable over dcl(ā_{i}) ∩ dcl(ā_{j}). But ā_{0} | ⌣ ā_{1} implies ā_{i} | ⌣ ā_{j} by indiscernibility. Thus σ_{i}(H^{c}) is a non-trivial group of finite SU-rank definable over acl(∅). If r ∈ S_{1}(acl(∅)) is a generic type for σ_{i}(H^{c}), then SU(r) = SU(σ_{i}(H^{c})) = SU(H^{c}) = SU(H) > n. This contradicts the maximal choice of n, as r does not fork over ∅. □

References

[1] Walter Baur, Gregory Cherlin and Angus Macintyre, Totally categorical groups and rings, J. Algebra, 57 (1979), 407–440.

[2] George M. Bergman and Hendrik W. Lenstra, Jr., Subgroups close to normal subgroups, J. Algebra, 127 (1989), 80–97.

[3] Gregory Cherlin, Leo Harrington and Alistair Lachlan, ℵ_{0}-categorical ℵ_{0}-stable structures, Ann. Pure Appl. Logic, 28 (1985), 103–135.

[4] David M. Evans, ℵ_{0}-categorical groups of finite rank, notes (1998).

[5] Ehud Hrushovski, Simplicity and the Lascar group, notes (1997).

[6] H. D. Macpherson, Absolutely ubiquitous structures and ℵ_{0}-categorical groups, Quart. J. Math. Oxford (2), 39 (1988), 483–500.

[7] Bruno Poizat, Groupes Stables, Nur al mantiq wal marifah, 1988.

[8] Massoud Pourmahdian, notes 1998.

[9] G. Schlichting, Operationen mit periodischen Stabilisatoren, Arch. Math. (Basel), 34 (1980), 97–99.

[10] Frank O. Wagner, Groups in simple theories, submitted to the J. Symb. Logic (1997).

Frank O Wagner, Mathematical Institute, University of Oxford, 24–29 St Giles’, Oxford OX1 3LB, UK

 E-mail address: wagner@maths.ox.ac.uk