Past Junior Topology and Group Theory Seminar

18 October 2017
16:00
Abstract

I will present a gentle introduction to the theory of conformal dimension, focusing on its applications to the boundaries of hyperbolic groups, and the difficulty of classifying groups whose boundaries have conformal dimension 1.

  • Junior Topology and Group Theory Seminar
26 April 2017
16:00
Abstract
We give a construction of a boundary (the Morse boundary) which can be assigned to any proper geodesic metric space and which is rigid, in the sense that a quasi-isometry of spaces induces a homeomorphism of boundaries. To obtain a more workable invariant than the homeomorphism type, I will introduce the metric Morse boundary and discuss notions of capacity and conformal dimensions of the metric Morse boundary. I will then demonstrate that these dimensions give useful invariants of relatively hyperbolic and mapping class groups. This is joint work with Matthew Cordes (Technion).
  • Junior Topology and Group Theory Seminar
1 March 2017
16:00
Benjamin Barrett
Abstract
Inspired by the theory of JSJ decomposition for 3-manifolds, one can define the JSJ decomposition of a group as a maximal canonical way of cutting it up into simpler pieces using amalgamated products and HNN extensions. If the group in question has some sort of non-positive curvature property then one can define a boundary at infinity for the group, which captures its large scale geometry. The JSJ decomposition of the group is then reflected in the treelike structure of the boundary. In this talk I will discuss this connection in the case of hyperbolic groups and explain some of the ideas used in its proof by Brian Bowditch.
  • Junior Topology and Group Theory Seminar
15 February 2017
16:00
Alex Margolis
Abstract
Asymptotic dimension is a large-scale analogue of Lebesgue covering dimension. I will give a gentle introduction to asymptotic dimension, prove some basic propeties and give some applications to group theory. I will then define coarse homology and explain how when defined, virtual cohomological dimension gives a lower bound on asymptotic dimension.
  • Junior Topology and Group Theory Seminar
1 December 2016
16:00
Nicolaus Heuer
Abstract

I will compare features of (classical) cohomology theory of groups to the rather exotic features of bounded (or continuous bounded) cohomology of groups.
Besides giving concrete examples I will state classical cohomological tools/features and see how (if) they survive in the case of bounded cohomology. Such will include the Mayer-Vietoris sequence, the transfer map, resolutions, classifying spaces, the universal coefficient theorem, the cup product, vanishing results, cohomological dimension and relation to extensions. 
Finally I will discuss their connection to each other via the comparison map.

  • Junior Topology and Group Theory Seminar
25 November 2016
15:00
Stefano Riolo
Abstract

By gluing copies of a deforming polytope, we describe some deformations of complete, finite-volume hyperbolic cone four-manifolds. Despite the fact that hyperbolic lattices are locally rigid in dimension greater than three (Garland-Raghunathan), we see a four-dimensional analogue of Thurston's hyperbolic Dehn filling: a path of cone-manifolds $M_t$ interpolating between two cusped hyperbolic four-manifolds $M_0$ and $M_1$.

This is a joint work with Bruno Martelli.

  • Junior Topology and Group Theory Seminar
2 November 2016
16:00
Alex Margolis
Abstract

Stallings' theorem states that a finitely generated group splits over a finite subgroup if and only if it has more than one end. As a consequence of this, group splittings over finite subgroups are invariant under quasi-isometry. I will discuss a generalisation of Stallings' theorem which shows that under suitable hypotheses, group splittings over classes of infinite groups, namely coarse $PD_n$ groups, are also invariant under quasi-isometry.

  • Junior Topology and Group Theory Seminar
26 October 2016
16:00
Claudio Llosa Isenrich
Abstract

A Kähler group is a group which can be realised as fundamental group of a compact Kähler manifold. I shall begin by explaining why such groups are not arbitrary and then address Delzant-Gromov's question of which subgroups of direct products of surface groups are Kähler. Work of Bridson, Howie, Miller and Short reduces this to the case of subgroups which are not of type $\mathcal{F}_r$ for some $r$. We will give a new construction producing Kähler groups with exotic finiteness properties by mapping products of closed Riemann surfaces onto an elliptic curve. We will then explain how this construction can be generalised to higher dimensions. This talk is independent of last weeks talk on Kähler groups and all relevant notions will be explained.

  • Junior Topology and Group Theory Seminar

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