**Birkbeck, University of London, Wednesday December 9th, 2015**

All talks will be held in room 402 in Birkbeck, University of London

If you wish to attend, please contact b.fairbairn [at] bbk.ac.uk

13:30-14:30 Robert Kropholler (Oxford) Bestvina-Brady Morse Theory and Subgroups of Hyperbolic Groups

I will introduce the key concepts from Bestvina-Brady Morse theory showing how this can be used to give groups with exotic finiteness properties. I will then show how this is used to construct finitely presented subgroups of hyperbolic groups which are not hyperbolic. Finally, I will discuss the limitations of the methods used in this construction.

14:30-15:30 Charles Cox (Southampton) An introduction to the R_{∞} property

There has been much work on the R_{∞} property: finding families of groups which satisfy it, and groups which do not. In this talk we will consider the following question. Let *G* be a group which acts faithfully on an infinite set *X*. Let *FSym(X)* denote the group of all finitely supported permutations of *X* (those which move finitely many points). Then does <*G, FSym(X)*> have the R_{∞} property? There will be some nice arguments using cycle type along the way.

15:30-16:00 Coffee

16:00-17:00 Alina Vdovina (Newcastle) Quadratic equations in hyperbolic groups are NP-complete

Using the geometry of closed surfaces we solve equations in hyperbolic and toral relatively hyperbolic graphs. We also show that the tasks of solving quadratic equations in hyperbolic and toral relatively hyperbolic groups are NP-complete. The main results are based joint work with O. Kharlampovich, A. Mohajeri and A. Taam.