On the linearity of finitely presented groups and its connection to the existence of good geometric actions

 The question of whether a finitely generated or presented group is linear has been looked at in various contexts, as has the existence of well behaved geometric actions of such groups by isometries on metric spaces although there is no single universally accepted definition of what such a well behaved action should be.
 
However at first sight there appears little connection between these two concepts, for instance one might consider some infinite simple groups or Baumslag-Solitar groups to make this point. In this talk we will first look at such groups to illustrate known results, before giving examples and evidence for the implication "linear implies good behaviour geometrically" if we use the appropriate notion of linearity.