Large free subgroups of automorphism groups of ultrahomogeneous spaces

Szymon Głąb; Filip Strobin

Colloquium Mathematicae (2015)

  • Volume: 140, Issue: 2, page 279-295
  • ISSN: 0010-1354

Abstract

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We consider the following notion of largeness for subgroups of S ∞ . A group G is large if it contains a free subgroup on generators. We give a necessary condition for a countable structure A to have a large group Aut(A) of automorphisms. It turns out that any countable free subgroup of S ∞ can be extended to a large free subgroup of S ∞ , and, under Martin’s Axiom, any free subgroup of S ∞ of cardinality less than can also be extended to a large free subgroup of S ∞ . Finally, if Gₙ are countable groups, then either ∏ n ∈ ℕ G ₙ is large, or it does not contain any free subgroup on uncountably many generators.

How to cite

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Szymon Głąb, and Filip Strobin. "Large free subgroups of automorphism groups of ultrahomogeneous spaces." Colloquium Mathematicae 140.2 (2015): 279-295. <http://eudml.org/doc/284204>.

@article{SzymonGłąb2015,
abstract = {We consider the following notion of largeness for subgroups of $S_\{∞\}$. A group G is large if it contains a free subgroup on generators. We give a necessary condition for a countable structure A to have a large group Aut(A) of automorphisms. It turns out that any countable free subgroup of $S_\{∞\}$ can be extended to a large free subgroup of $S_\{∞\}$, and, under Martin’s Axiom, any free subgroup of $S_\{∞\}$ of cardinality less than can also be extended to a large free subgroup of $S_\{∞\}$. Finally, if Gₙ are countable groups, then either $∏_\{n∈ℕ\} Gₙ$ is large, or it does not contain any free subgroup on uncountably many generators.},
author = {Szymon Głąb, Filip Strobin},
journal = {Colloquium Mathematicae},
keywords = {ultrahomogeneus structures; large substructures; free groups},
language = {eng},
number = {2},
pages = {279-295},
title = {Large free subgroups of automorphism groups of ultrahomogeneous spaces},
url = {http://eudml.org/doc/284204},
volume = {140},
year = {2015},
}

TY - JOUR
AU - Szymon Głąb
AU - Filip Strobin
TI - Large free subgroups of automorphism groups of ultrahomogeneous spaces
JO - Colloquium Mathematicae
PY - 2015
VL - 140
IS - 2
SP - 279
EP - 295
AB - We consider the following notion of largeness for subgroups of $S_{∞}$. A group G is large if it contains a free subgroup on generators. We give a necessary condition for a countable structure A to have a large group Aut(A) of automorphisms. It turns out that any countable free subgroup of $S_{∞}$ can be extended to a large free subgroup of $S_{∞}$, and, under Martin’s Axiom, any free subgroup of $S_{∞}$ of cardinality less than can also be extended to a large free subgroup of $S_{∞}$. Finally, if Gₙ are countable groups, then either $∏_{n∈ℕ} Gₙ$ is large, or it does not contain any free subgroup on uncountably many generators.
LA - eng
KW - ultrahomogeneus structures; large substructures; free groups
UR - http://eudml.org/doc/284204
ER -

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