Leon Pernak: Introduction to hyperbolic towers and generalized Fraïssé limits
Thursday, 13.01.2022 11:00 im Raum via Zoom
A long standing question in both model theory and group theory has been
whether all non-abelian free groups are elementarily equivalent. The
question was famously asked by Tarski around 1945 and answered
positively by Z. Sela and, independently, by O. Kharlampovich and A.
Myasnikov in 2006. In pursuit of the question, Sela studied what he
called (hyperbolic) towers. This concept was further investigated by C.
Perin and R. Sklinos, among others, who worked out multiple details and
further applications of the concept.
The central justification for studying hyperbolic towers is the
following result due to Sela:
If G is a non-abelian torsion-free hyperbolic group and a hyperbolic
tower over some non-abelian subgroup H, then H is elementarily embedded
in G.
And its converse, due to Perin:
If a torsion-free hyperbolic group H is elementarily embedded into some
torsion-free hyperbolic group G, then G is a hyperbolic tower over H.
The definition of hyperbolic towers relies heavily on concepts in
geometric group theory. Therefore we will quickly discuss fundamental
groups of complexes and graphs of groups. The latter provide a tool to
decompose groups into amalgamated products and HNN-extensions, known as
Bass-Serre theory. Towers then consist of multiple layers of such
decompositions with certain additional properties. We will work our way
through the definitions along multiple examples and state the main
results.
If time permits, we will also discuss a generalization of classical
Fraïssé limits, which was used by Kharlampovich-Myasnikov and later by
Guirardel-Levitt-Sklinos to provide a homogeneous group in which all
non-abelian free groups (Kharlampovich-Myasnikov) or more generally,
all elementarily equivalent torsion-free groups
(Guirardel-Levitt-Sklinos) embed elementarily.
Angelegt am 01.12.2021 von Martina Pfeifer
Geändert am 10.12.2021 von Martina Pfeifer
[Edit | Vorlage]