2 results
ON THE AUTOMORPHISM GROUP OF THE UNIVERSAL HOMOGENEOUS MEET-TREE
- Part of
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- Journal:
- The Journal of Symbolic Logic / Volume 86 / Issue 4 / December 2021
- Published online by Cambridge University Press:
- 01 February 2021, pp. 1508-1540
- Print publication:
- December 2021
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- Article
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COHERENT EXTENSION OF PARTIAL AUTOMORPHISMS, FREE AMALGAMATION AND AUTOMORPHISM GROUPS
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- Journal:
- The Journal of Symbolic Logic / Volume 85 / Issue 1 / March 2020
- Published online by Cambridge University Press:
- 06 May 2019, pp. 199-223
- Print publication:
- March 2020
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- Article
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We give strengthened versions of the Herwig–Lascar and Hodkinson–Otto extension theorems for partial automorphisms of finite structures. Such strengthenings yield several combinatorial and group-theoretic consequences for homogeneous structures. For instance, we establish a coherent form of the extension property for partial automorphisms for certain Fraïssé classes. We deduce from these results that the isometry group of the rational Urysohn space, the automorphism group of the Fraïssé limit of any Fraïssé class that is the class of all
${\cal F}$-free structures (in the Herwig–Lascar sense), and the automorphism group of any free homogeneous structure over a finite relational language all contain a dense locally finite subgroup. We also show that any free homogeneous structure admits ample generics.
