Open Problems in Group Theory and Topology

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OPEN PROBLEMS IN GROUP THEORY AND TOPOLOGY Igor Mineyev. Math 595, second half of Fall 2018 (October 22 - December 12), MWF 3pm. www.math.uiuc.edu/~mineyev/class/18f/595/ There is a deep relationship between groups, topological spaces, and metric spaces. Studying one helps studying another. Groups can be viewed as geometric objects, to any topological space one can associate its fundamental group, group presentations lead to cell complexes, metric spaces can be studied using group actions, etc. Geometric group theory is the area of mathematics investigating such relations. In this course I will concentrate on multiple (and very difficult) open problems that empha- size such relations. If you can solve any one of those, turn it to me as homework. Here are open-ended themes that I plan to address in the course. (1) 1-complexes: Cayley graphs of groups, free groups and their subgorups, Nielsen transfor- mations, the Hanna Neumann conjecture (solved), generalizations of HNC, ... (2) Free groups and free products: one-relator groups, Magnus’ Freiheitssatz and its generalizations, the Kervaire conjecture, ... (3) 2-complexes: group presentations, (highly nontrivial) presentations of the trivial group, the Andrews-Curtis conjecture, the Grigorchuk-Kurchanov conjecture, aspherical 2-complexes, the Whitehead conjecture, the Bestvina-Brady construction, ... (4) Geometric methods: hyperbolic groups and spaces, the boundary at infinity of a hyperbolic group, Cannon’s conjecture, ... (5) 3-manifolds: the Poincaréconjecture (from a group-theoretic perspective), Stallings’ ap- proach (“How not to prove the Poincaréconjecture”), surfaces in 3-manifods, ... If you are interested in exploring these realms deeper, see the references to books on this class website. This course will only run if sufficiently many students sign up..

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Arxiv:1006.1489V2 [Math.GT] 8 Aug 2010 Ril.Ias Rfie Rmraigtesre Rils[14 Articles Survey the Reading from Profited Also I Article
Pure and Applied Mathematics Quarterly Volume 8, Number 1 (Special Issue: In honor of F. Thomas Farrell and Lowell E. Jones, Part 1 of 2 ) 1—14, 2012 The Work of Tom Farrell and Lowell Jones in Topology and Geometry James F. Davis∗ Tom Farrell and Lowell Jones caused a paradigm shift in high-dimensional topology, away from the view that high-dimensional topology was, at its core, an algebraic subject, to the current view that geometry, dynamics, and analysis, as well as algebra, are key for classifying manifolds whose fundamental group is infinite. Their collaboration produced about fifty papers over a twenty-five year period. In this tribute for the special issue of Pure and Applied Mathematics Quarterly in their honor, I will survey some of the impact of their joint work and mention briefly their individual contributions – they have written about one hundred non-joint papers. 1 Setting the stage arXiv:1006.1489v2 [math.GT] 8 Aug 2010 In order to indicate the Farrell–Jones shift, it is necessary to describe the situation before the onset of their collaboration. This is intimidating – during the period of twenty-five years starting in the early fifties, manifold theory was perhaps the most active and dynamic area of mathematics. Any narrative will have omissions and be non-linear. Manifold theory deals with the classification of ∗I thank Shmuel Weinberger and Tom Farrell for their helpful comments on a draft of this article. I also profited from reading the survey articles [14] and [4]. 2 James F. Davis manifolds. There is an existence question – when is there a closed manifold within a particular homotopy type, and a uniqueness question, what is the classification of manifolds within a homotopy type? The fifties were the foundational decade of manifold theory.
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