Simplicity of some automorphism groups

Abstract Let M be a countably infinite first order relational structure which is homogeneous in the sense of Fraisse. We show, under the assumption that the class of finite substructures of M has the free amalgamation property, along with the assumption that Aut ( M ) is transitive on M but not equal to Sym ( M ) , that Aut ( M ) is a simple group. This generalises results of Truss, Rubin and others. The proof uses the Polish group structure of the automorphism group and generalises to certain other homogeneous structures, with prospects for further application.

[1]  C. Ward Henson,et al.  Countable homogeneous relational structures and ℵ0-categorical theories , 1972, Journal of Symbolic Logic.

[2]  Chen C. Chang,et al.  Model Theory: Third Edition (Dover Books On Mathematics) By C.C. Chang;H. Jerome Keisler;Mathematics , 1966 .

[3]  Peter J. Cameron.  A census of infinite distance-transitive graphs , 1998, Discret. Math..

[4]  Ehud Hrushovski,et al.  A New Strongly Minimal Set , 1993, Ann. Pure Appl. Log..

[5]  A. Kechris.  Classical descriptive set theory , 1987 .

[6]  Edmund F. Robertson,et al.  Infinite simple permutation groups - a survey , 1991 .

[7]  Shashi M. Srivastava,et al.  A Course on Borel Sets , 1998, Graduate texts in mathematics.

[8]  Wacław Sierpiński,et al.  Sur un espace métrique séparable universel , 1945 .

[9]  D. G. Higman.  Intersection matrices for finite permutation groups , 1967 .

[10]  P. J. Cameron,et al.  Some isometry groups of the Urysohn space , 2004, Ann. Pure Appl. Log..

[11]  P. Cameron,et al.  Oligomorphic permutation groups , 1990 .

[12]  Frank O. Wagner.  Relational structures and dimensions , 1994 .

[13]  Daniel Lascar.  Les beaux automorphismes , 1991, Arch. Math. Log..

[14]  Daniel Lascar.  Les Automorphismes d'un Ensemble Fortement Minimal , 1992, J. Symb. Log..

[15]  John K. Truss,et al.  The group of almost automorphisms of the countable universal graph , 1989 .

[16]  Manfred Droste,et al.  Structure of partially ordered sets with transitive automorphism groups , 1985 .

[17]  G. Cherlin,et al.  The Classification of Countable Homogeneous Directed Graphs and Countable Homogeneous N-Tournaments , 1998 .

[18]  T. S. Gardener.  Infinite Dimensional Classical Groups , 1995 .

[19]  Bernhard Herwig.  Extending partial isomorphisms for the small index property of many ω-categorical structures , 1998 .

[20]  Dugald Macpherson,et al.  A survey of homogeneous structures , 2011, Discret. Math..

[21]  K. Tent.  Free Polygons, Twin Trees, and CAT(1)-Spaces , 2011 .

[22]  A. Kechris,et al.  Turbulence, amalgamation, and generic automorphisms of homogeneous structures , 2004, math/0409567.

[23]  Matatyahu Rubin,et al.  Automorphism groups of countable highly homogeneous partially ordered sets , 1993 .

[24]  John K. Truss,et al.  The group of the countable universal graph , 1985, Mathematical Proceedings of the Cambridge Philosophical Society.

[25]  A. Ivanov.  Automorphisms of Homogeneous Structures , 2005, Notre Dame J. Formal Log..

[26]  Nellie Clarke Brown.  Trees , 1896, Savage Dreams.

[27]  Manfred Droste,et al.  Automorphism Groups of Infinite Semilinear Orders (I) , 1989 .

[28]  J. K. Truss.  The automorphism group of the random graph: four conjugates good, three conjugates better , 2003, Discret. Math..