Abstract, Bass, 9700634 The project will address three areas of geometric group theory. The first concerns group actions on trees, particularly tree lattices, i.e. discrete groups of automorphisms of locally finite trees with finite volume quotients. This theory is a discrete analogue of the theory of Fuchsian groups acting on the upper half plane. The second area is the representation theory of finitely generated groups, and particularly the structure of rigid groups, i.e. those with only finitely many irreducible representations in each dimension. When also linear, these are conjecturally of arithmetic type. The third area concerns algebraic group actions on affine space, and, in particular, flat families of representations of reductive groups, and the extent to which these are globally equivariant vector bundles. The project is an investigation of certain groups of symmetries of various kinds of geometric structures. In the first instance these are infinite trees, ( i.e. connected graphs without closed circuits), and our symmetry groups are discrete, and act with say finitely many orbits. In the second setting, we study finitely generated groups via their matrix representations, particularly for groups that have "very few" of these. In the last case we study Lie groups acting algebraically, but possibly non linearly, on Euclidean spaces, and try to determine when certain such actions are "approximately linear."
