Cusps of hermitian locally symmetric spaces

This thesis explores the geometry at infinity for certain hermitian locally symmetric spaces. Let Gamma < SU(r + 1, r) be a maximal nonuniform arithmetic lattice determined by automorphisms of a hermitian form on k^{2 r + 1}, where k is an imaginary quadratic field. We give a formula for the number of cusps of X / Gamma, where X is the hermitian symmetric space on which SU(r + 1, r) acts. If r > 1 and 2 r + 1 is prime, this completely determines the number of cusps for minimal finite volume orbifolds with X-geometry, and there are only finitely many commensurability classes of noncompact finite volume quotients of X containing a one-cusped orbifold. In the case r = 1, which corresponds to the complex hyperbolic plane, we show that this holds for any N: there are only finitely many commensurability classes of arithmetic lattices in SU(2, 1) which contain an N-cusped orbifold.