D. Groves: Towards the elementary theory of relatively hyperbolic groups

Abstract: We will discuss some work which attempts to extend Sela's work on the elementary theory from torsion-free hyperbolic groups to relatively hyperbolic groups. In particular, we will focus on torsion-free hyperbolic groups with abelian parabolics, where such an extension is possible. The key is to build an appropriate R-tree, and to prove an analogue of the shortening argument. Time permitting, we will discuss the difficulty of extending this approach to arbitrary relatively hyperbolic groups.