Fra¨ ıss´ e Limits, Hrushovski Property and Generic Automorphisms Shixiao Liu acid@pku.edu.cn June 2017 Abstract This paper is a survey on the Fra¨ ıss´ e limits of the classes of fi- nite graphs, finite rational metric spaces and finite K n -free graphs. Hrushovski showed a property concerning extending partial isomor- phisms for finite graphs, and this property turns out to yield results for the Fra¨ ıss´ e limit of the class of finite graphs. Similar properties were proven for the other two classes, which yielded analogue results concerning the generic automorphisms of their Fra¨ ıss´ e limits. 1 Introduction The beginning of the story we’re going to tell dated back to the 1950s. Roland Fra¨ ıss´ e published an important paper [3], where he established a method proven to be fruitful in constructing countably infinite homogeneous structures. Following Hodges’s suggestion in [7], we now call this method Fra¨ ıss´ e construction. Among all the structures we obtain from Fra¨ ıss´ e con- struction, the first one that might draw most mathematician’s attention was the Fra¨ ıss´ e limit of the class of finite graphs. In 1963, Erd˝ os and R´ enyi [2] proved that, if a countable graph is chosen at random by selecting edges independently with probability 1 / 2, then the resulting graph is isomorphic to the Fra¨ ıss´ e limit of the class of finite graphs with probability 1. This 1
is why we call this graph the random graph . (The graph is also known as Rado’s graph, since it was Rado who gave the first concrete construction of it.) In 1992, Hrushovski published his famous paper [9], where he proved that, every finite graph can be extended to a finite supergraph, such that all the partial isomorphisms of the original graph extend to automorphisms of the supergraph. It is shown that, this result, together with other results about generic automorphisms, yields several properties concerning the au- tomorphism group of the random graph, for example, the small index prop- erty as proven by Hodges, Hodkinson, Lascar and Shelah in [8]. Kechris and Rosendal showed later in [10] that, their proof also applies to other Fra¨ ıss´ e limits, when certain conditions are met for the Fra¨ ıss´ e classes. Inter- esting enough, in many cases the conditions Kechris and Rosendal came up with can be implied from results analogue to Hrushovski’s theorem for finite graphs. This leads us to the concern whether it’s feasible to extend partial isomorphisms within other Fra¨ ıss´ e classes. Herwig and Lascar established a method in [6] which would help. All these results have brought about the manufacturing of properties in a large number of seemingly different struc- tures. To summarize their work, we will consider in this article several most basic Fra¨ ıss´ e classes and prove parallel results for each of them. This paper mainly consists of three parts. In the first part, we introduce Fra¨ ıss´ e construction and define the structures with which we concern in the following sections. In the second part, we give proofs of results that allow us to extend arbitrary partial isomorphism, namely the Hrushovski property, for the Fra¨ ıss´ e classes we mentioned in the first section. We also answer one of Hrushovski’s questions he raised in [9]. In the last part, we focus on the automorphism groups of the Fra¨ ıss´ e limits and apply the Hrushovski property for each Fra¨ ıss´ e class to prove the existence of generic automorphisms. These proofs based on more recent results are much shorter than their original proofs. We also prove another result that shows the 2
generic automorphisms are conjugate to their powers for these Fra¨ ıss´ e limits. In this article we need some basic notions from first order logic, especially model theory. We recall that, a language L is a collection of function sym- bols, relational symbols and constant symbols. A language is relational if it contains no function symbols and constants. An L -structure M is an un- derlying set M with interpretation of each symbol in L . Two L -structures M and N are isomorphic if there is a bijection f : M → N preserving the interpretation of all symbols in L . A function g is a partial isomorphism between M and N if it is an isomorphism between substructures of M and N . For an L -structure M and an L -formula φ , the notion that M satis- fies φ (denoted by M | = φ ) is defined by induction on the complexity of φ . For a more detailed introduction to these notions, the reader may refer to textbooks of model theory, such as Hodges [7] and Marker [12]. 2 Fra¨ ıss´ e Limits e proved that, for a class K of struc- In his classic paper [3], Roland Fra¨ ıss´ tures with some certain properties, there exists a structure that contains isomorphic copies of every structure in K . In order to state his theorem, we first introduce some definitions. Definition 2.1. Let K be a class of structures. 1. K has the hereditary property (HP for short), if for all A ∈ K and all substructures B of A , we have B ∈ K . 2. K has the joint embedding property (JEP for short), if for all A, B ∈ K , there is C ∈ K such that both A and B can be embedded in C . 3. K has the amalgamation property (AP for short), if for all A, B and C in K with embeddings f 1 : A → B , f 2 : A → C , there is D ∈ K with embeddings g 1 : B → D , g 2 : C → D such that g 1 ◦ f 1 = g 2 ◦ f 2 . 3
Definition 2.2. Let L be a relational language and M be an L -structure. The age of M is the class K of all finite structures that can be embedded in M . Definition 2.3. Let L be a relational language and M be an L -structure. We call M ultrahomogeneous if every isomorphism between two finite sub- structures of M extends to an automorphism of M . With the definitions above, we can now state Fra¨ ıss´ e’s theorem: Theorem 2.4 (Fra¨ ıss´ e’s Theorem, [7, Thm 7.1.2]) . Let L be a countable language and let K be a non-empty finite or countable class of finite L - structures which has HP, JEP and AP. Then there is an L -structure M , unique up to isomorphism, such that M has cardinality at most ω , the class K is the age of M and M is ultrahomogeneous. Following Hodges’s terminology, a class which satisfies the conditions of the theorem is called a Fra¨ ıss´ e class , with the uniquely determined L - structure M to be its Fra¨ ıss´ e limit . We now introduce the three main structures we’re concerned with in this article. Example 2.5 (The random graph) . Let K be the class of all finite graphs, where the language of graphs contains only one binary relation symbol E and no function symbols or constants. For every finite graph G , the universe of the structure is the set of all vertices in G ; for every two vertices v 1 , v 2 ∈ G , v 1 and v 2 are connected iff G | = E ( v 1 , v 2 ) ∧ E ( v 2 , v 1 ). It’s obvious that K has HP, JEP and AP, and thus the class K has a unique Fra¨ ıss´ e limit, which is called the random graph R . The word “random” here is due to Erd˝ os and R´ enyi, as introduced before. Example 2.6 (The Henson graph) . For each positive integer n , let K n be the class of all finite graphs which do not include K n , i.e. , the complete graph of size n , as an induced subgraph. The language we work with and its interpretation are the same as that in the previous example. It’s obvious 4
that HP and JEP hold for K n . To check AP, we take graph A, B, C ∈ K n with embeddings f 1 : A → B , f 2 : A → C . Without loss of generality, we may assume A ⊆ B and A ⊆ C . Let D be the set of B ∪ C and define two vertices to be adjacent in D iff they’re adjacent in either B or C . Since neither B nor C contains K n as an induced subgraph, D is K n -free. Thus AP holds for the class K n . Since this Fra¨ ıss´ e class was first introduced and e limit of the class K n the Henson studied by Henson in [4], we call the Fra¨ ıss´ graph H n . Example 2.7 (The rational Urysohn space) . Let K be the class of all finite rational metric spaces, i.e. , the class of finite metric spaces that take dis- tances in positive rational numbers. The language L we work with contains a relation symbol R r for each positive rational number r and no function symbols or constants. For every finite rational metric space X , the universe of the structure is the set of all points in X ; for every two distinct points a, b ∈ X , we have d ( a, b ) = r iff X | = R r ( a, b ) ∧ R r ( b, a ). Again, it’s obvious that K has HP and JEP. To check AP, we take finite rational metric spaces A, B and C with embeddings f 1 : A → B , f 2 : A → C . Without loss of generality, we may assume A ⊆ B and A ⊆ C . Let D be the set B ∪ C . We define a metric on D by preserving the metric on B or C when a, b ∈ B or a, b ∈ C and setting d D ( a, b ) to be the minimum of d B ( a, x ) + d C ( x, b ) for all x ∈ A when a ∈ B \ A and b ∈ C \ A . It’s straight forward to check that ( D, d D ) is a finite rational metric space, and therefore AP holds for the class K . Due to Urysohn [17], we call the Fra¨ ıss´ e limit of K the rational Urysohn space QU . 3 Hrushovski Property In his famous work [9], Hrushovski proved the following result: Theorem 3.1 (Hrushovski, [9]) . Given a finite graph X , there is a finite graph Z such that X is an induced subgraph of Z and every partial isomor- 5
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