Invited Talks Abstracts

This talk will be addressed to a general mathematical audience. The contents are a consequence of the close connection between combinatorial group theory and the topology of surfaces, where the fundamental groups are either one relator groups or free groups. Consider the free Z-module generated by the set of directed free homotopy classes of closed curves on a orientable surface. Goldman proved that on this Z-module there exists a Lie algebra structure, obtained by combining the geometric intersection of curves with the usual loop product of curves at intersection points. Since free homotopy classes of curves on a surface are in one to one correspondence with conjugacy classes of the fundamental group, this implies for each surface the existence of a Lie algebra structure on the Z-module generated by the conjugacy classes of the surface group. We will give a definition of this Lie algebra and discuss several of its properties. When a free homotopy class of curves is simple, i.e. has a representative with no self-intersections, one can write the fundamental group of the surface as an amalgamated free product or an HNN extension of certain subgroups. Using this fact, we will give a combinatorial description of the Goldman Lie bracket of a simple class with any other free homotopy class. We will use this description to prove that there is no cancellation and consequently the number of terms of the bracket, counted with multiplicity, is the minimal intersection number of theses two free homotopy classes. When the surface has non-empty boundary, there is a combinatorial presentation of the entire Goldman Lie bracket. Using this presentation we find an algorithm to count the minimal number of self-intersections of a free homotopy class of curves. There are open problems related to the characterization of which conjugacy classes contain simple closed curves. Namely, certain ”commutator words” constructed using a word which has a simple representative should not be conjugate. There is also the problem of determining the centers of these Lie algebras. Natasha Dobrinen (University of Denver) The tree property Abstract: This is joint work with Sy-David Friedman. The well-known Koenig’s Lemma states that every finitely branching tree of countably infinite height contains an infinite path through the tree. However, the natural generalization of Koenig’s Lemma to trees of uncountable height breaks down. Precisely, Aronszajn constructed a tree of height ω1 (the first uncountable cardinal) such that every level of the tree is countable, yet the tree contains no branch going all the way through it. Such a tree is called an ω1-Aronszajn tree. When looking at larger uncountable cardinals, the picture becomes even more complicated. For any regular cardinal κ ≥ א2, the non-existence of κ-Aronszajn trees requires axioms in addition to those of ZFC (the standard axioms of set theory, and much of mathematics in general). In this talk, we will investigate when κ-Aronszajn trees exist and when they do not. In particular, we find the equiconsistency strength of there being no κ-Aronszajn trees when κ is a measurable cardinal. Olga Kharlampovich (McGill University) Elementary theory of a free group and related questions This is joint work with Sy-David Friedman. The well-known Koenig’s Lemma states that every finitely branching tree of countably infinite height contains an infinite path through the tree. However, the natural generalization of Koenig’s Lemma to trees of uncountable height breaks down. Precisely, Aronszajn constructed a tree of height ω1 (the first uncountable cardinal) such that every level of the tree is countable, yet the tree contains no branch going all the way through it. Such a tree is called an ω1-Aronszajn tree. When looking at larger uncountable cardinals, the picture becomes even more complicated. For any regular cardinal κ ≥ א2, the non-existence of κ-Aronszajn trees requires axioms in addition to those of ZFC (the standard axioms of set theory, and much of mathematics in general). In this talk, we will investigate when κ-Aronszajn trees exist and when they do not. In particular, we find the equiconsistency strength of there being no κ-Aronszajn trees when κ is a measurable cardinal. Olga Kharlampovich (McGill University) Elementary theory of a free group and related questions Abstract: I will briefly outline some of the key points (see below) in our proof with A. Myasnikov of the Tarski conjectures about the elementary theory of a free group and outline some applications of methods and techniques. These conjectures stated that the elementary theory of non-abelian free groups of different ranks coinside and that this common theory is decidable. The first conjecture was independently proved by Sela. The key points are: 1. Development of the algebraic geometry over groups in several papers by Baumslag, Myasnikov, Remeslennikov and myself; 2. The theory of fully residually free groups (limit groups) and a simple algebraic description of them, embedding of I will briefly outline some of the key points (see below) in our proof with A. Myasnikov of the Tarski conjectures about the elementary theory of a free group and outline some applications of methods and techniques. These conjectures stated that the elementary theory of non-abelian free groups of different ranks coinside and that this common theory is decidable. The first conjecture was independently proved by Sela. The key points are: 1. Development of the algebraic geometry over groups in several papers by Baumslag, Myasnikov, Remeslennikov and myself; 2. The theory of fully residually free groups (limit groups) and a simple algebraic description of them, embedding of