Oligomorphic permutation groups: growth rates and algebras Peter J. - PDF document

Oligomorphic permutation groups: growth rates and algebras Peter J. Cameron p.j.cameron@qmul.ac.uk Gregynog Mathematics Colloquium22 May 2007 The definition Examples, 2 Consider the group S r acting on the disjoint union Let G be a permutation group on an infinite set Ω . Then G has a natural induced action on the set of all of r copies of X . n -tuples of elements of Ω , or on the set of n -tuples of distinct elements of Ω , or on the set of n -element • F n ( S r ) = r n ; subsets of Ω . It is easy to see that if there are only finitely many orbits on one of these sets, then the • f n ( S r ) = ( n + r − 1 r − 1 ) . same is true for the others. Consider S r acting on Ω r . Then F ∗ n ( S r ) = B ( n ) r . We say that G is oligomorphic if it has only finitely many orbits on Ω n for all natural numbers n . From this we can find F n ( S r ) by inversion: We denote the number of orbits on all n -tuples, n resp. n -tuples of distinct elements, n -sets, by F ∗ s ( n , k ) F ∗ n ( G ) , ∑ F n ( G ) = k ( G ) F n ( G ) , f n ( G ) respectively. k = 1 for any oligomorphic group G , where s ( n , k ) is the Examples, 1 signed Stirling number of the second kind. Let S be the symmetric group on an infinite set X . Then S is oligomorphic and For A 2 acting on Q 2 , f n ( A 2 ) is the number of zero- • F n ( S ) = f n ( S ) = 1, one matrices (of unspecified size) with n ones and no rows or columns of zeros. • F ∗ n ( S ) = B ( n ) , the n th Bell number (the number of partitions of a set of size n . Let A = Aut ( Q , < ) , the group of order-preserving Examples, 3 permutations of Q . Then A is oligomorphic and Let G = S Wr S , the wreath product of two copies of S . Then F n ( G ) = B ( n ) and f n ( G ) = p ( n ) , the • f n ( A ) = 1; number of partitions of n . • F n ( A ) = n !; Let G = S 2 Wr A , where S 2 is the symmetric group • F ∗ n ( A ) is the number of preorders of an n -set. of degree 2. Then f n ( G ) is the n th Fibonacci number . 1

Growth of ( f n ( G )) , 1 Examples, 4 Several things are known about the behaviour of There is a unique countable random graph R : that the sequence ( f n ( G )) : is, if we choose a countable graph at random (edges independent with probability 1 2 , then with probabil- • it is non-decreasing; ity 1 it is isomorphic to R . • either it grows like a polynomial (that is, an k ≤ f n ( G ) ≤ bn k for some a , b > 0 and k ∈ N ), or it • R is universal , that is, it contains every finite or grows faster than any polynomial; countable graph as an induced subgraph; • if G is primitive (that is, it preserves no non- • R is homogeneous , that is, any isomorphism be- trivial equivalence relation on Ω ), then either tween finite induced subgraphs of R can be ex- f n ( G ) = 1 for all n , or f n ( G ) grows at least ex- tended to an automorphism of R . ponentially; • if G is highly homogeneous (that is, if f n ( G ) = 1 If G = Aut ( R ) , then F n ( G ) and f n ( G ) are the num- for all n ), then either there is a linear or circular bers of labelled and unlabelled graphs on n vertices. order on Ω preserved or reversed by G , or G is highly transitive (that is, F n ( G ) = 1 for all n ). Connection with model theory, 1 • There is no upper bound on the growth rate of If a set of sentences in a first-order language has ( f n ( G )) . an infinite model, then it has arbitrarily large infi- nite models. In other words, we cannot specify the Growth of ( f n ( G )) , 2 cardinality of an infinite structure by first-order ax- Examples suggest that much more is true. For any ioms. reasonable growth rate, appropriate limits should exist: Cantor proved that a countable dense total order without endpoints is isomorphic to Q . Apart from • for polynomial growth of degree k , countability, the conditions in this theorem are all lim ( f n ( G ) / n k ) should exist; first-order sentences. • for fractional exponential growth (like exp ( n c ) ), What other structures can be specified by count- lim ( log log f n ( G ) / log n ) should exist; ability and first-order axioms? Such structures are lim ( log f n ( G ) / n ) • for exponential growth, called countably categorical . should exist; and so on. Connection with model theory, 2 I do not know how to prove any of these things; In 1959, the following result was proved indepen- and I do not know how to formulate a general con- dently by Engeler, Ryll-Nardzewski and Svenonius: jecture. Theorem 1. A countable structure M over a first-order A Ramsey-type theorem language is countably categorical if and only if Aut ( M ) is oligomorphic. Theorem 2. Let X be an infinite set, and suppose that the n-element subsets of Ω are coloured with r different In fact, more is true: the types over the theory of M colours (all of which are used). Then there is an ordering are all realised in M , and the sets of n -tuples which ( c 1 , . . . , c r ) of the colours, and infinite subsets Y 1 , . . . , Y r realise the n -types are precisely the orbits of Aut ( M ) of X, such that, for i = 1, . . . , r, the set Y i contains an on M n . n-set of colour c i but none of colour c j for j > i. 2

If G is oligomorphic, then the dimension of V G The existence of Y 1 is the classical theorem of Ram- n is f n ( G ) , and so the Hilbert series of the algebra A [ G ] sey. is the ordinary generating function of the sequence There is a finite version of the theorem, and so ( f n ( G )) . there are corresponding ‘Ramsey numbers’. But What properties does this algebra have? very little is known about them! Note that it is not usually finitely generated since the growth of ( f n ( G )) is polynomial only in special Monotonicity cases. Corollary 3. The sequence ( f n ( G )) is non-decreasing. A non-zero-divisor Let e be the constant function in V 1 with value 1. Of course, e lies in A [ G ] for any permutation group Proof. Let r = f n ( G ) , and colour the n -subsets with G . r colours according to the orbits. Then by the Theo- rem, there exists an ( n + 1 ) -set containing a set of Theorem 4. The element e is not a zero-divisor in A . colour c i but none of colour c j for j > i . These ( n + 1 ) -sets all lie in different orbits; so f n + 1 ( G ) ≥ This theorem gives another proof of the mono- r . tonicity of ( f n ( G )) . For multiplication by e is a monomorphism from V G n to V G n + 1 , and so f n + 1 ( G ) = There is also an algebraic proof of this corollary. We’ll discuss this later. dim v G n + 1 ≥ dim V G n = f n ( G ) . A graded algebra, 1 An integral domain Let ( Ω If G has a finite orbit ∆ , then any function whose n ) denote the set of n -subsets of Ω , and V n support is contained in ∆ is nilpotent. the vector space of functions from ( Ω n ) to C . The converse, a long-standing conjecture, has re- We make A = � n ≥ 0 V n into an algebra by defin- cently been proved by Maurice Pouzet: ing, for f ∈ V n , g ∈ V m , the product f g ∈ V n + m by Theorem 5. If G has no finite orbits on Ω , then A [ G ] is ( f g )( K ) = ∑ f ( M ) g ( K \ M ) an integral domain. M ∈ ( K m ) Consequences for K ∈ ( Ω m + n ) , and extending linearly. Pouzet’s Theorem has a consequence for the growth rate: A is a commutative and associative graded alge- bra over C , sometimes referred to as the reduced inci- Theorem 6. If G is oligomorphic, then dence algebra of finite subsets of Ω . f m + n ( G ) ≥ f m ( G ) + f n ( G ) − 1. Proof. Multiplication maps V G m ⊗ V G n into V G m + n ; by A graded algebra, 2 Pouzet’s result, it is injective on the projective Segre Now let G be a permutation group on Ω , and let variety, and a little dimension theory gets the result. V G n denote the set of fixed points of G in V n . Put V G � A [ G ] = n , It seems very likely that better understanding of n ≥ 0 the algebra A [ G ] would have further implications a graded subalgebra of A . for growth rate. 3

Brief sketch of the proof Let F be a family of subsets of Ω . A subset T is transversal to F if it intersects each member of F . The transversality of F is the minimum cardinality of a transversal. A lemma due to Peter Neumann shows that, if G has no finite orbits on Ω , then any orbit of G On fi- nite sets has infinite transversality. Pouzet shows that, if f ∈ V m and g ∈ V n satisfy f g = 0, then the transversality of supp ( f ) ∪ supp ( g ) is finite, and is bounded by a function of m and n . (Here supp ( f ) denotes the support of f .) These two results clearly conflict with each other. Comments Here is Pouzet’s theorem again: Theorem 7. If f ∈ V m and g ∈ V n satisfy f g = 0 , then the transversality of supp ( f ) ∪ supp ( g ) is finite, and is bounded by a function of m and n. The proof of this makes it clear that it is another kind of ‘Ramsey theorem’. If τ ( m , n ) denotes the smallest t such that the transversality is at most t , then we have the interesting problem of finding τ ( m , n ) . Pouzet shows that τ ( m , n ) ≥ ( m + 1 )( n + 1 ) − 1. On the other hand, the upper bounds coming from his proof are really astronomical! 4