turkmath.org

The discovery of non euclidean geometry in the early nineteenth century had shaken the beliefs and conjectures of more than two thousand years and changed the picture we had for mathematics, physics and even philosophy. Lobachevsky and Bolyai independently around 1830 discovered hyperbolic geometry. A notable distinguish feature of hyperbolic geometry is its negative curvature in a way that the sum of angles of a triangle is less than π. Gromov much later in 1987 introduced hyperbolic groups which are groups acting “nicely” on hyperbolic spaces, or equivalently finitely generated groups whose Cayley graphs are “negatively curved”. Main examples are free groups and almost all surface groups. The fascinating subject of hyperbolic groups touches on many mathematical disciplines such as geometric group theory, low dimensional topology and combinatorial group theory. It is connected to model theory through a question of Tarski.

Tarski asked around 1946 whether non abelian free groups have the same first order theory. This question proved extremely hard to answer and only after more than fifty years in 2001 Sela and Kharlampovich-Myasnikov answered it positively. Both works are voluminous and have not been fully absorbed yet. The great novelty of the methods and the depth of the needed results have made it hard to streamline any of the proofs. Despite the difficulties there is some considerable progress in the understanding of the first-order theory of “the free group” and consequently first-order theories of hyperbolic groups from the scopes of basic model theory, Shelah’s classification theory and geometric stability. In this talk I will survey what is known about these theories and what are the main open questions.