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Martina Pfeifer

Leon Pernak: Introduction to hyperbolic towers and generalized Fraïssé limits

Thursday, 13.01.2022 11:00 im Raum via Zoom

Mathematik und Informatik

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 Wednesday, 01.12.2021 09:47 von Martina Pfeifer
Geändert am Friday, 10.12.2021 11:00 von Martina Pfeifer
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