Groups and Geometry in the South East meeting Event

Schedule of talks and abstracts:

• 1:30-2:30 Javier Aramayona (Madrid): On the abelianization of (pure) big mapping class groups.
  A classical theorem of Powell asserts that the mapping class group of an orientable surface of finite topological type and genus at least three has trivial abelianization. The first part of the talk will be devoted to explaining a proof of this result, as well as discussing the remaining low-genus cases.
  We will then show that, in stark contrast, mapping class groups of infinite-type surfaces can have infinite abelianization. More concretely, we will explain how to construct non-trivial integer-valued homomorphisms from mapping class groups of infinite-genus surfaces. Further, we will give a description the first integral cohomology group of pure mapping class groups in terms of the first homology of the underlying surface. This is joint work with Priyam Patel and Nick Vlamis.
• 2:40-3:40 Robert Kropholler (Tufts): Almost finitely presented subgroups of hyperbolic groups.
  Hyperbolic groups form a well understood class of groups. However, the subgroups of hyperbolic groups can be much wilder. One might hope that by imposing extra conditions upon subgroups they become easier to understand. A positive result of Gersten shows that if G is hyperbolic and has cohomological dimension 2 and H is a subgroup of type FP_2, then H is hyperbolic. In particular, H is finitely presented.
  I will detail work showing that this phenomenon is special to dimension 2 by constructing examples of subgroups of hyperbolic groups which are of type FP_2 but not finitely presented.
• 3:40-4:00 TEA
• 4:00-5:00 Alina Vdovina (Newcastle): Ramanujan cubical complexes as higher-dimensional expanders.
  Ramanujan graphs were first considered by Lubotzky, Phillips, Sarnak to get graphs with optimal spectral properties. In our days the theory of expander graphs and, in particular, Ramanujan graphs is well developed, but the questions is what is the best definition of a higher-dimensional expander is still wide open. There are several approaches, suggested by Gromov, Lubotzky, Alon and others, but the cubical complexes were not much investigated from this point of view. In this talk I will give new explicit examples of cubical Ramanujan complexes and discuss possible developments.

For more information, see the GGSE website.