Integrality and arithmeticity of co-compact lattice corresponding to certain complex two-ball quotients of Picard number one
Let \(G\) be a semi-simple Lie group of non-compact type and \(K\) be a maximal compact subgroup of \(G\), then \(G/K\) is a symmetric space of noncompact type. Let \(\Gamma\) be a co-compact lattice in \(G\). In the case when \(G/K\) has real rank greater than or equal to 2, \(\Gamma\) has to be an arithmetic lattice by Margulis' arithmeticity theorem. If \(G/K\) is either a quaternionic hyperbolic space or the Cayley hyperbolic plane, \(\Gamma\) is arithmetic by the results of Corlette and Gromov-Schoen. On the other hand, in the case of \(G/K\) being a real or a complex hyperbolic space non-arithmetic lattices were constructed by Gromov and Piatetski-Shapiro, and Deligne and Mostow. In these cases one might ask whether there are additional conditions that ensure the arithmeticity of the lattices. For lattices in \(PU(2,1)\) (acting on the complex hyperbolic plane), such a condition in terms of cohomology groups was conjectured by Rogawski. In Part A of the main theorem the author of the paper under review proves that lattices in \(PU(2,1)\) that satisfy the conditions conjectured by Rogawski are integral, which is slightly weaker than being arithmetic. In Parts B and C of the main theorem the author shows that under additional conditions on either the canonical line bundle or the absence of an immersed totally geodesic curve the lattices are arithmetic.
scientific article; zbMATH DE number 2148244