Georgia Institute of Technology College of Sciences

Any knot in $S^3$ may be reduced to a slice knot by making some crossing changes. Indeed, this slice knot can be taken to be the unknot. We show that the same is true of knots in homology spheres, at least topologically. Something more complicated is true smoothly, as not every homology sphere bounds a smooth simply connected homology ball. We prove that a knot in a homology sphere is null-homotopic in a homology ball if and only if that knot can be reduced to the unknot by a sequence of concordances and crossing changes. We will show that there exist knot in a homology sphere which cannot be reduced to the unknot by any such sequence. As a consequence, there are knots in homology spheres which are not concordant to those examples produced by Levine in 2016 and Hom-Lidman-Levine in 2018.

We show that the three-dimensional homology cobordism group admits an infinite-rank summand. It was previously known that the homology cobordism group contains an infinite-rank subgroup and a Z-summand. Our proof relies on the involutive Heegaard Floer package of Hendricks-Manolescu and Hendricks-Manolescu-Zemke. This is joint work with I. Dai, M. Stoffregen, and L. Truong.

The notion of an acylindrically hyperbolic group was introduced by Osin as a generalization of non-elementary hyperbolic and relative hyperbolic groups. Ex- amples of acylindrically hyperbolic groups can be found in mapping class groups, outer automorphism groups of free groups, 3-manifold groups, etc. Interesting properties of acylindrically hyperbolic groups can be proved by applying techniques such as Monod-Shalom rigidity theory, group theoretic Dehn filling, and small cancellation theory. We have recently shown that non-elementary convergence groups are acylindrically hyperbolic. This result opens the door for applications of the theory of acylindrically hyperbolic groups to non-elementary convergence groups. In addition, we recovered a result of Yang which says a finitely generated group whose Floyd boundary has at least 3 points is acylindrically hyperbolic.

I will discuss knot concordances in 3-manifolds. In particular I will talk about knot concordances of knots in the free homotopy class of S^1 x {pt} in S^1 x S^2. It turns out, we can use some of these concordances to construct Akbulut-Ruberman type exotic 4-manifolds. As a consequence, at the end of the talk we will see absolutely exotic Stein pair of 4-manifolds. This is joint work with Selman Akbulut.