The conjugacy problem for the automorphism group of the random graph

Coskey, Samuel; Ellis, P.J.; Schneider, Scott

doi:10.1007/s00153-010-0210-y

Cited by 4 publications

(11 citation statements)

References 8 publications

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“…In the article , we showed with Schneider that the conjugacy problem for

Aut (Γ)

is Borel complete. The next result gives a streamlined version of the argument from , and at the same time generalizes it to work for

Aut ({normalΓ}_{n})

too. Theorem Let

n \geq 3

.…”

Section: Undirected Graphsmentioning

confidence: 99%

“…Lachlan and Woodrow classified the countably infinite homogeneous undirected graphs as follows: (a)for

m, n \leq \infty

and either m or n infinite, the graph

m \cdot K_{n}

consisting of m many disjoint copies of

K_{n}

(§ ); (b)the generic undirected graph, also known as the random graph (cf. ); (c)for

n < \infty

, the generic

K_{n}

‐free graph (§ ); and (d)graph complements of each of these (they have the same automorphism group). …”

Section: Undirected Graphsmentioning

confidence: 99%

“…Returning to conjugacy, after the results summarized in we formulated a conjecture that the conjugacy problem for automorphism groups of countable homogeneous structures is always either smooth (for “trivial” homogeneous structures like

double-struckN

with no relations) or Borel complete (for “complicated” homogeneous structures like the random graph). After studying further examples, we observe that this pattern mostly holds, even though we found an exception in Theorem .…”

Section: Introductionmentioning

confidence: 99%

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The conjugacy problem for automorphism groups of countable homogeneous structures

Coskey

Ellis

2016

Math. Log. Quart.

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“…In the article , we showed with Schneider that the conjugacy problem for

Aut (Γ)

is Borel complete. The next result gives a streamlined version of the argument from , and at the same time generalizes it to work for

Aut ({normalΓ}_{n})

too. Theorem Let

n \geq 3

.…”

Section: Undirected Graphsmentioning

confidence: 99%

“…Lachlan and Woodrow classified the countably infinite homogeneous undirected graphs as follows: (a)for

m, n \leq \infty

and either m or n infinite, the graph

m \cdot K_{n}

consisting of m many disjoint copies of

K_{n}

(§ ); (b)the generic undirected graph, also known as the random graph (cf. ); (c)for

n < \infty

, the generic

K_{n}

‐free graph (§ ); and (d)graph complements of each of these (they have the same automorphism group). …”

Section: Undirected Graphsmentioning

confidence: 99%

double-struckN

Section: Introductionmentioning

confidence: 99%

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The conjugacy problem for automorphism groups of countable homogeneous structures

Coskey

Ellis

2016

Math. Log. Quart.

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“…The automorphism groups of countable saturated structures have been the subject of much study, and in many cases the conjugacy problem is known to be Borel complete. For example, the conjugacy problem for the automorphism group of the rational linear ordering (Q, <), the random graph, and the atomless Boolean algebra are all known to be Borel complete (for a discussion of these results, see [1]). It is shown in [8] that if M is a countable recursively saturated model of PA, then By a theorem of Smoryński [13], a cut I of a countable recursively saturated model of PA is of the form I fix (f) for some f ∈ Aut(M ) if and only if it is closed under exponentiation.…”

Section: Theorem 43 Suppose That C Is a Class Of Countable Models Smentioning

confidence: 99%

“…is called the space of countable models of Θ. 1 Studying the classification problem for countable Θ-models now amounts to studying the isomorphism equivalence relation ∼ = Θ on X Θ . Now, if E, F are (not necessarily Borel) equivalence relations on the standard Borel spaces X, Y , then we say that E is Borel reducible to F (written E ≤ B F ) iff there exists a Borel function f: X → Y such that…”

mentioning

confidence: 99%

The Complexity of Classification Problems for Models of Arithmetic

Coskey¹,

Kossak²

2010

Bull. symb. log

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ABSTRACT. We observe that the classification problem for countable models of arithmetic is Borel complete. On the other hand, the classification problems for finitely generated models of arithmetic and for recursively saturated models of arithmetic are Borel; we investigate the precise complexity of each of these. Finally, we show that the classification problem for pairs of recursively saturated models and for automorphisms of a fixed recursively saturated model are Borel complete.

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Conjugacy for homogeneous ordered graphs

Coskey¹,

Ellis²

2018

Arch. Math. Logic

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A countable relational structure M is said to be homogeneous if every finite partial automorphism of M extends to an automorphism of M. The automorphism groups of homogeneous structures can have striking properties, see for instance [Mac11, Chapter 5]. In recent work [CES11, CE16, CE17], we investigated the conjugacy classification problem for the automorphism groups of a variety of well-studied homogeneous structures such as graphs and digraphs. We found that with few exceptions, the conjugacy problem was of the maximum conceivable complexity.In order to say what we mean by "maximum conceivable complexity", we very briefly recall the Borel complexity theory of equivalence relations. We refer the reader to [Gao09] for more on this broadly applicable area of descriptive set theory. If E, F are equivalence relations on standard Borel spaces X, Y, we say E is Borel reducible to F if there exists a Borel function f :. If Y is a space countable structures with isomorphism relation ∼ = Y , we say that ∼ = Y is Borel complete if for every space X of countable structures with isomorphism relation ∼ = X we have that ∼ = X is Borel reducible to ∼ = Y . For example the isomorphism relations on the classes of countable graphs, tournaments, and linear orders are all Borel complete.Since conjugacy of automorphisms f of a fixed structure M is equivalent to isomorphism of expanded structures (M, f ), it makes sense to ask whether the conjugacy classification of automorphisms of M is Borel complete. For most of the homogeneous structures M that we considered in our recent work, we showed that the conjugacy classification of automorphisms of M is Borel complete.In this note we extend this family of results to the automorphisms of homogeneous ordered digraphs. A structure M is said to be ordered if one of the symbols of M is a binary relation < that satisfies the axioms of a linear order. Recently, Cherlin [Che18] classified the countable homogeneous ordered graphs. We will use this classification to establish our main result: for every ordered graph G the conjugacy classification of automorphisms of G is Borel complete. CONJUGACY FOR HOMOGENEOUS ORDERED GRAPHS 2Our proof strategy involves showing that each homogeneous ordered digraph possesses a very strong property called the ABAP. Before defining this property, we first recall that if M is a homogeneous structure, then one commonly studies the class K of finite structures isomorphic to a substructure of M and the class K ω of countable structures isomorphic to a substructure of M. These classes hold a variety of extension and amalgamation properties. (We refer the reader to [Hod93, Chapter 7] for a starting point on amalgamation classes and related properties.) In our proofs, we will show that for each homogeneous ordered graph G, the corresponding class K ω satisfies a very strong kind of extension property. We then show that this property can be used to define a Borel reduction from a known Borel complete relation to the conjugacy relation on Aut(G).We now define the extension...

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The conjugacy problem for the automorphism group of the random graph

Abstract: ABSTRACT. We prove that the conjugacy problem for the automorphism group of the random graph is Borel complete, and discuss the analogous problem for some other countably categorical structures.

Cited by 4 publications

References 8 publications

The conjugacy problem for automorphism groups of countable homogeneous structures

The conjugacy problem for automorphism groups of countable homogeneous structures

The Complexity of Classification Problems for Models of Arithmetic

Conjugacy for homogeneous ordered graphs

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