FRAISSE STRUCTURES AND THEIR AUTOMORPHISM GROUPS
(PART 2)
MIKE COHEN
Definition. Let F be a structure with countable universe F . Let Aut F denote the
group of all automorphisms of F, endowed with the topology of pointwise convergence
on F , where F is viewed as a discrete space. This topology makes Aut F into a Polish
group.
Theorem. Let F be a structure with universe ω. Then Aut F is a closed subgroup of
S∞ .
Proof. Aut F is obviously a subgroup, so it suffices to show that Aut F is closed in
S∞ . We will show that the complement X = S∞ − Aut F is open. Let L be the
signature for which F is a structure. If π ∈ X, then either there exists an n-ary
relation symbol R ∈ L and a tuple (x1 , ..., xn ) ∈ ω n such that
(x1 , ..., xn ) ∈ RF and (π(x1 ), ..., π(xn )) ∈
/ RF , or (x1 , ..., xn ) ∈
/ RF and
F
(π(x1 ), ..., π(xn )) ∈ R ,
or else there exists an n-ary function symbol f ∈ L and a tuple (x1 , ..., xn ) ∈ ω n such
that
f F (π(x1 ), ..., π(xn )) 6= π(f F (x1 , ..., xn )).
In either case, the open set U = {σ ∈ S∞ : σ(x1 ) = π(x1 ), ..., σ(xn ) = π(xn )} is a
neighborhood of π which is disjoint from Aut F.
Definition. Let G be a subgroup of S∞ . We associate a structure FG to G as follows:
For each n ∈ ω, let ω n /G denote the set of all distinct G-orbits in ω n . Let LG be a
signature consisting of relation symbols Rn,o , where n ∈ ω and o ∈ ω n /G, where the
FG
arity of each Rn,o is n. The universe of the structure FG is ω, and we define each Rn,o
by the rule
FG
(x1 , ..., xn ) ∈ Rn,o
↔ (x1 , ..., xn ) ∈ o.
1
2
MIKE COHEN
FG
= o! We call FG the canonical structure associated
In other words, we set Rn,o
to G.
Remark. It is obvious that every g ∈ G will induce an automorphism of FG . Thus we
have G ≤ Aut F.
Theorem. Aut FG is the closure of G in S∞ .
Proof. Since we already know Aut FG is closed, it suffices to show that G is dense in
Aut FG . So let π ∈ Aut FG , let x = (x1 , ..., xn ) ∈ ω n , and consider the basic open
neighborhood U of π defined by
U = {σ ∈ S∞ : σ(xi ) = π(xi ), i = 1, ..., n}.
Let o ∈ ω n /G be the unique orbit for which x ∈ o. Then since π ∈ Aut FG , we
have πx ∈ o. But by definition we have o = G · x, and hence there exists a g ∈ G for
which gx = πx. Hence g ∈ U , and the proof is finished.
Proposition. FG is a Fraisse structure.
Proof. FG is obviously countably infinite, and since it is a relational structure, it is
also locally finite. So we need only verify that FG is ultrahomogeneous.
So suppose A, B v FG are finite and p : A ,→ B is an isomorphism. Let
A = {x1 , ..., xn } ⊆ ω and B = {y1 , ..., yn } ⊆ ω be the universes of A and B respectively, enumerated in such a way that p(xi ) = yi for all i = 1, .., n. Let o ∈ ω n /G
FG
B
FG
A
∩ Bn = o ∩ Bn,
= Rn,o
∩ An = o ∩ An and Rn,o
= Rn,o
be arbitrary. Since we have Rn,o
it follows that
(x1 , ..., xn ) ∈ o ↔ (x1 , ..., xn ) ∈ o ∩ An
A
↔ (x1 , ..., xn ) ∈ Ro,n
B
↔ (y1 , ..., yn ) ∈ Ro,n
↔ (y1 , ..., yn ) ∈ o ∩ B n
↔ (y1 , ..., yn ) ∈ o.
So (x1 , ..., xn ) and (y1 , ..., yn ) belong to the same G-orbit, i.e. there exists a g ∈ G
with (g(x1 ), ..., g(xn )) = (y1 , ..., yn ) = (p(x1 ), ..., p(xn )). So g ∈ Aut FG and g extends
p; thus FG is ultrahomogeneous.
FRAISSE STRUCTURES AND THEIR AUTOMORPHISM GROUPS (PART 2)
3
Corollary. The automorphism groups of Fraisse structures on ω are exactly the closed
subgroups of S∞ .
Corollary. Every automorphism group of a Fraisse structure is topologically isomorphic to a closed subgroup of S∞ . In particular, Aut Q, Aut R, Aut D, Aut F∞ ,
Aut B∞ , and Aut UQ may all be regarded as closed subgroups of S∞ .