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Series: Geometry Topology Seminar

We use Manolescu's Pin(2)-equivariant Floer homology to study homology cobordisms among Seifert spaces. In particular, we will show that the subgroup of the homology cobordism group generated by Seifert spaces admits a \mathbb{Z}^\infty summand. This is joint work with Irving Dai.

Series: Geometry Topology Seminar

We give "visual descriptions" of cut points and non-parabolic cut pairs in the Bowditch boundary of a relatively hyperbolic right-angled Coxeter group. We also prove necessary and sufficient conditions for a relatively hyperbolic right-angled Coxeter group whose defining graph has a planar flag complex with minimal peripheral structure to have the Sierpinski carpet or the 2-sphere as its Bowditch boundary. We apply these results to the problem of quasi-isometry classification of right-angled Coxeter groups. Additionally, we study right-angled Coxeter groups with isolated flats whose $\CAT(0)$ boundaries are Menger curve. This is a joint work with Matthew Haulmark and Hoang Thanh Nguyen.

Series: Geometry Topology Seminar

Let M be a closed hyperbolic 3-manifold with a fibered face \sigma of the unit ball of the Thurston norm on H_2(M). If M satisfies a certain condition related to Agol’s veering triangulations, we construct a taut branched surface in M spanning \sigma. This partially answers a 1986 question of Oertel, and extends an earlier partial answer due to Mosher. I will not assume knowledge of the Thurston norm, branched surfaces, or veering triangulations.

Series: Geometry Topology Seminar

Series: Geometry Topology Seminar

Series: Geometry Topology Seminar

The mapping class group of the plane minus a Cantor set naturally appears in many dynamical contexts, including group actions on surfaces, the study of groups of homeomorphisms on a Cantor set, and complex dynamics. In this talk, I will present the 'ray graph', which is a Gromov-hyperbolic graph on which this big mapping class group acts by isometries (it is an equivalent of the curve graph for this surface of infinite topological type). If time allows, I will give a description of the Gromov-boundary of the ray graph in terms of long rays in the plane minus a Cantor set. This involves joint work with Alden Walker.

Series: Geometry Topology Seminar

We will give different topological very simple statements that seem not to have been noticed, although they are of the level
of Brouwer’s fixed point theorem. The main result is: Let F be a compact subset of the manifold M. Assume g:F->M is a continuous map which is the identity on the boundary (or frontier) of F, then the image g(F) contains either F or M\F.

Series: Geometry Topology Seminar

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Series: Geometry Topology Seminar

Series: Geometry Topology Seminar

There has been a recent interest in studying surfaces of infinite type, i.e. surfaces with infinitely-generated fundamental groups. In this talk, we will focus on their mapping class groups, often called big mapping class groups. In contrast to the finite-type case, there are many open questions regarding the basic algebraic and topological properties of big mapping class groups. I will discuss several such questions and provide some answers. In particular, I will discuss automorphism groups of mapping class groups as well as relations between topological invariants of a surface and algebraic invariants of its mapping class group. The results in the talk are based on recent joint work with Priyam Patel and ongoing joint work with Javier Aramayona and Priyam Patel.