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

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Symmetries have played an important role in sciences and artsand are described in terms of group theory. Two important types ofgroups are discrete groups and Lie groups.Lie groups are closely related to homogeneous spaces.An important class of homogeneous spaces consists ofsymmetric spaces, a very distinguished class of special Riemannianmanifolds. Examples of symmetric spaces include the Euclidean spacesand the spheres in them. Another type of symmetric spacesis given by hyperbolic spaces, in which there are infinitelymany nonparallel lines which do not intersect with each other.Discrete groups acting on symmetric spaces give rise to locallysymmetric spaces; for example, surfaces with constant curvatureare locally symmetric spaces.The interplay between the topology, geometry and group theoryof discrete groups, Lie groups and locally symmetric spaceshas been intensively studied in mathematics.In this proposal, the PI proposes to study the Novikov conjecturesfor arithmetic groups using the large scale geometry andcompactifications of symmetric spaces of noncompact type.Specifically, an important invariant of the asymptotic geometryis the asymptotic dimension, the finiteness of which is closelyrelated to the Novikov conjectures. For a torsion free arithmeticsubgroup of a semisimple algebraic group,the partial Borel-Serre compactificationof the associated symmetric space is the universal coveringof the classifying space of the arithmetic group. For applicationsto the Novikov conjectures, we need a large compactification of thepartial Borel-Serre compactification. To study S-arithmetic subgroups,a generalization of arithmetic groups, we also need compactificationsof Bruhat-Tits buildings. Compactifications of symmetricspaces and buildings are also important for other purposes.