Quotients of the complex ball by discrete groups

Frances Kirwan, Ronnie Lee and Steven Howard Weintraub

Vol. 130 (1987), No. 1, 115–141

DOI: 10.2140/pjm.1987.130.115

Abstract

In this paper we systematically study varieties Q(μ), which are compactifications of the space Q of distinct points in (P¹)^r given by a sequence of “weights” μ, and which for certain μ are also compactification of the quotient of the complex r-ball by discrete subgroups Γ(μ) of PU(r,1), as discovered by Deligne and Mostow.

We obtain a wealth of topological information about the spaces Q(μ) and their desingularizations Q^∗(μ). In some cases we can completely describe them. Otherwise, we obtain computations of Betti numbers and Hodge numbers. As applications we determine the L²-cohomology and in many cases the (ordinary) rational cohomology of the groups Γ(μ).

Mathematical Subject Classification 2000

Primary: 32J05

Secondary: 11F25, 14J15, 22E40, 32M10, 32C40

Milestones

Received: 12 February 1986

Published: 1 November 1987