Elementary embeddings in torsion-free hyperbolic groups

Chloé Perin

Annales scientifiques de l'École Normale Supérieure (2011)

  • Volume: 44, Issue: 4, page 631-681
  • ISSN: 0012-9593

Abstract

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We describe first-order logic elementary embeddings in a torsion-free hyperbolic group in terms of Sela’s hyperbolic towers. Thus, if H embeds elementarily in a torsion free hyperbolic group Γ , we show that the group Γ can be obtained by successive amalgamations of groups of surfaces with boundary to a free product of  H with some free group and groups of closed surfaces. This gives as a corollary that an elementary subgroup of a finitely generated free group is a free factor. We also consider the special case where Γ is the fundamental groups of a closed hyperbolic surface. The techniques used to obtain this description are mostly geometric, as for example actions on real or simplicial trees, or the existence of JSJ splittings. We also rely on the existence of factor sets, a result used in the construction of Makanin-Razborov diagrams for torsion-free hyperbolic groups.

How to cite

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Perin, Chloé. "Elementary embeddings in torsion-free hyperbolic groups." Annales scientifiques de l'École Normale Supérieure 44.4 (2011): 631-681. <http://eudml.org/doc/272248>.

@article{Perin2011,
abstract = {We describe first-order logic elementary embeddings in a torsion-free hyperbolic group in terms of Sela’s hyperbolic towers. Thus, if $H$ embeds elementarily in a torsion free hyperbolic group $\Gamma $, we show that the group $\Gamma $ can be obtained by successive amalgamations of groups of surfaces with boundary to a free product of $H$ with some free group and groups of closed surfaces. This gives as a corollary that an elementary subgroup of a finitely generated free group is a free factor. We also consider the special case where $\Gamma $ is the fundamental groups of a closed hyperbolic surface. The techniques used to obtain this description are mostly geometric, as for example actions on real or simplicial trees, or the existence of JSJ splittings. We also rely on the existence of factor sets, a result used in the construction of Makanin-Razborov diagrams for torsion-free hyperbolic groups.},
author = {Perin, Chloé},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {geometric group theory; first-order logic; trees (graph theory); free groups; Tarski problem; Sela’s hyperbolic towers; elementary substructures},
language = {eng},
number = {4},
pages = {631-681},
publisher = {Société mathématique de France},
title = {Elementary embeddings in torsion-free hyperbolic groups},
url = {http://eudml.org/doc/272248},
volume = {44},
year = {2011},
}

TY - JOUR
AU - Perin, Chloé
TI - Elementary embeddings in torsion-free hyperbolic groups
JO - Annales scientifiques de l'École Normale Supérieure
PY - 2011
PB - Société mathématique de France
VL - 44
IS - 4
SP - 631
EP - 681
AB - We describe first-order logic elementary embeddings in a torsion-free hyperbolic group in terms of Sela’s hyperbolic towers. Thus, if $H$ embeds elementarily in a torsion free hyperbolic group $\Gamma $, we show that the group $\Gamma $ can be obtained by successive amalgamations of groups of surfaces with boundary to a free product of $H$ with some free group and groups of closed surfaces. This gives as a corollary that an elementary subgroup of a finitely generated free group is a free factor. We also consider the special case where $\Gamma $ is the fundamental groups of a closed hyperbolic surface. The techniques used to obtain this description are mostly geometric, as for example actions on real or simplicial trees, or the existence of JSJ splittings. We also rely on the existence of factor sets, a result used in the construction of Makanin-Razborov diagrams for torsion-free hyperbolic groups.
LA - eng
KW - geometric group theory; first-order logic; trees (graph theory); free groups; Tarski problem; Sela’s hyperbolic towers; elementary substructures
UR - http://eudml.org/doc/272248
ER -

References

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