Large Scale Geometry and Compactifications of Arithmetic Groups, Symmetric Spaces and Buildings

Regents Of The University Of Michigan - Ann Arbor, Ann Arbor MI

Investigators

Abstract

Symmetries have played an important role in sciences and arts and are described in terms of group theory. Two important types of groups are discrete groups and Lie groups. Lie groups are closely related to homogeneous spaces. An important class of homogeneous spaces consists of symmetric spaces, a very distinguished class of special Riemannian manifolds. Examples of symmetric spaces include the Euclidean spaces and the spheres in them. Another type of symmetric spaces is given by hyperbolic spaces, in which there are infinitely many nonparallel lines which do not intersect with each other. Discrete groups acting on symmetric spaces give rise to locally symmetric spaces; for example, surfaces with constant curvature are locally symmetric spaces. The interplay between the topology, geometry and group theory of discrete groups, Lie groups and locally symmetric spaces has been intensively studied in mathematics. In this proposal, the PI proposes to study the Novikov conjectures for arithmetic groups using the large scale geometry and compactifications of symmetric spaces of noncompact type. Specifically, an important invariant of the asymptotic geometry is the asymptotic dimension, the finiteness of which is closely related to the Novikov conjectures. For a torsion free arithmetic subgroup of a semisimple algebraic group, the partial Borel-Serre compactification of the associated symmetric space is the universal covering of the classifying space of the arithmetic group. For applications to the Novikov conjectures, we need a large compactification of the partial Borel-Serre compactification. To study S-arithmetic subgroups, a generalization of arithmetic groups, we also need compactifications of Bruhat-Tits buildings. Compactifications of symmetric spaces and buildings are also important for other purposes.