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Feb. 5: Topology Seminar – Jens Harlander

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3 p.m. MB124. Jens Harlander of Boise State will be the speaker for this seminar.

Abstract: The Whitehead conjecture states that a subcomplex of an aspherical 2-complex is aspherical. This is a 2-dimensional version of the true statement “a subgraph of a cycle free graph is cycle free.” Whitehead put forth his conjecture in 1941, it arose in connection with the asphericity question for knot complements. In my talk I will adress a recent result of Hillman and Osin, which says: Let K be a subcomplex of an aspherical 2-complex and let G be its fundamental group. Assume that G maps onto the infinite cyclic group (a mild assumption). If the first L2-Betti number of K vanishes, then K is aspherical. If it does not vanish then G is acylindrically hyperbolic. In particular, G is SQ-universal, than means every countable group is a subgroup of a quotient of G.

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