A random group can be thought as of a finitely presented group given by a 'typical' group presentation, determined by drawing uniformly and randomly from a set of group presentations according to the precise rules of a model for the random group. In this project we plan to study, on the one hand, dynamical properties, like diffuseness (Bowditch), of random groups. On the other hand, we are interested in geometric and analytical properties of random groups, for example, the existence of a geometric action on a non positively curved cube complex, or, in contrast, Kazhdan's Property (T). More precisely, we want to explore these properties for random quotients of free products. Here the main tasks are (1) to adapt methods developed to study random groups in Gromov’s density model to the relatively geometric setting of free products, (2) to explore possible models for random quotients of free products. One motivation for this study is that it will provide a tool to study  ‘typical’ examples of relatively hyperbolic groups. While this is a worthwhile goal on its own, a general hope is that such a study helps to further develop the available methods and may thus be useful to develop new approaches to long-standing open questions in the field.

Research Team: MurphyKate Montee (Carleton College, Northfield), Markus Steenbock (U of Vienna)