76 research outputs found

    The conjugacy problem for automorphism groups of countable homogeneous structures

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    We consider the conjugacy problem for the automorphism groups of a number of countable homogeneous structures. In each case we find the precise complexity of the conjugacy relation in the sense of Borel reducibility

    Conjugacy for Homogeneous Ordered Graphs

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    We show that for any countable homogeneous ordered graph G, the conjugacy problem for automorphisms of G is Borel complete. In fact we establish that each such G satisfies a strong extension property called ABAP, which implies that the isomorphism relation on substructures of G is Borel reducible to the conjugacy relation on automorphisms of G

    Finite 33-connected homogeneous graphs

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    A finite graph \G is said to be {\em (G,3)(G,3)-((connected)) homogeneous} if every isomorphism between any two isomorphic (connected) subgraphs of order at most 33 extends to an automorphism gGg\in G of the graph, where GG is a group of automorphisms of the graph. In 1985, Cameron and Macpherson determined all finite (G,3)(G, 3)-homogeneous graphs. In this paper, we develop a method for characterising (G,3)(G,3)-connected homogeneous graphs. It is shown that for a finite (G,3)(G,3)-connected homogeneous graph \G=(V, E), either G_v^{\G(v)} is 22--transitive or G_v^{\G(v)} is of rank 33 and \G has girth 33, and that the class of finite (G,3)(G,3)-connected homogeneous graphs is closed under taking normal quotients. This leads us to study graphs where GG is quasiprimitive on VV. We determine the possible quasiprimitive types for GG in this case and give new constructions of examples for some possible types

    The random graph

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    Erd\H{o}s and R\'{e}nyi showed the paradoxical result that there is a unique (and highly symmetric) countably infinite random graph. This graph, and its automorphism group, form the subject of the present survey.Comment: Revised chapter for new edition of book "The Mathematics of Paul Erd\H{o}s

    Vertex-primitive digraphs with large fixity

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    The relative fixity of a digraph Γ\Gamma is defined as the ratio between the largest number of vertices fixed by a nontrivial automorphism of Γ\Gamma and the number of vertices of Γ\Gamma. We characterize the vertex-primitive digraphs whose relative fixity is at least 1/31/3, and we show that there are only finitely many vertex-primitive digraphs of bounded out-valency and relative fixity exceeding a positive constant.Comment: 16 page
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