Lattices In PU Research Articles

Complex hyperbolic orbifolds and hybrid lattices

A class of complex hyperbolic lattices in PU(2, 1) called the Deligne-Mostow lattices has been reinterpreted by Hirzebruch (see [1, 10] and [24]) in terms of line arrangements. They use branched covers over a suitable blow up of the complete quadrilateral arrangement of lines in mathbb {P}^2 to construct the complex hyperbolic surfaces over the orbifolds associated to the lattices. In [18] and [19], fundamental domains for these lattices have been built by Pasquinelli. Here we show how the fundamental domains can be interpreted in terms of line arrangements as above. This parallel is then applied in the following context. Wells in [25] shows that two of the Deligne-Mostow lattices in PU(2, 1) can be seen as hybrids of lattices in PU(1, 1). Here we show that he implicitly uses the line arrangement and we complete his analysis to all possible pairs of lines. In this way, we show that three more Deligne-Mostow lattices can be given as hybrids.

Residually Finite Lattices in PU(2,1)˜ and Fundamental Groups of Smooth Projective Surfaces

This paper studies residual finiteness of lattices in the universal cover of PU(2,1) and applications to the existence of smooth projective varieties with fundamental group a cocompact lattice in PU(2,1) or a finite covering of it. First, we prove that certain lattices in the universal cover of PU(2,1) are residually finite. To our knowledge, these are the first such examples. We then use residually finite central extensions of torsion-free lattices in PU(2,1) to construct smooth projective surfaces that are not birationally equivalent to a smooth compact ball quotient but whose fundamental group is a torsion-free cocompact lattice in PU(2,1).

Non-arithmetic monodromy of higher hypergeometric functions

We show that all the currently known non-arithmetic lattices in PU(2, 1) are monodromy groups of higher hypergeometric functions.

A new nonarithmetic lattice in PU(3,1)

We study the arithmeticity of the Couwenberg-Heckman-Looijenga lattices in PU(n,1), and show that they contain a non-arithmetic lattice in PU(3,1) which is not commensurable to the non-arithmetic Deligne-Mostow lattice in PU(3,1).

Fundamental domains and presentations for the Deligne–Mostow lattices with 2-fold symmetry

In this work we will build a fundamental domain for Deligne-Mostow lattices in PU(2,1) with 2-fold symmetry, which complete the whole list of Deligne-Mostow lattices in dimension 2. These lattices were introduced by Deligne and Mostow using monodromy of hypergeometric functions and have been reinterpreted by Thurston as authomorphisms on a sphere with cone singularities. Following his approach, Parker, Boadi and Parker, and Pasquinelli built a fundamental domain for the class of lattices with 3-fold symmetry, i.e. when three of five cone singularities have same cone angle. Here we extend this construction to the asymmetric case, where only two of the five cone points on the sphere have same cone angle, so to have a fundamental domain for each commensurability class of Deligne-Mostow lattices in PU(2,1).

Hybrid lattices and thin subgroups of Picard modular groups

We consider a certain hybridization construction which produces a subgroup of PU(n,1) from a pair of lattices in PU(n−1,1). Among the Picard modular groups PU(2,1,Od), we show that the hybrid of pairs of Fuchsian subgroups PU(1,1,Od) is a lattice when d=1 and d=7, and a geometrically infinite thin subgroup when d=3, that is an infinite-index subgroup with the same Zariski-closure as the full lattice.

Lattices in 𝑃𝑈(𝑛,1) that are not profinitely rigid

Using conjugation of Shimura varieties, we produce nonisomorphic, cocompact, torsion-free lattices in PU ⁡ ( n , 1 ) \operatorname {PU}(n,1) with isomorphic profinite completions for all n ≥ 2 n \ge 2 . This disproves a conjecture of D. Kazhdan and gives the first examples of nonisomorphic lattices in a semisimple Lie group of real rank one with isomorphic profinite completions, answering two questions of A. Reid.

Deligne-Mostow lattices with three fold symmetry and cone metrics on the sphere

Deligne and Mostow constructed a class of lattices inPU(2,1)PU(2,1)using monodromy of hypergeometric functions. Thurston reinterpreted them in terms of cone metrics on the sphere. In this spirit we construct a fundamental domain for the lattices with three fold symmetry in the list of Deligne and Mostow. This is a generalisation of the works of Boadi and Parker and gives a different interpretation of the fundamental domain constructed by Deraux, Falbel, and Paupert.

Mostow’s lattices and cone metrics on the sphere

Abstract In his seminal paper of 1980, Mostow constructed a family of lattices in PU(2, 1), the holomorphic isometry group of complex hyperbolic 2-space. In this paper, we use a description of these lattices given by Thurston in terms of cone metrics on the sphere, which is equivalent to Deligne and Mostow’s description of them using monodromy of hypergeometric functions. We give an explicit fundamental domain for some of Mostow’s lattices, specifically those with large phase shift. Our approach follows Parker’s approach of describing Livné’s lattices in terms of cone metrics on the sphere. The content of this paper is based on Boadi’s PhD thesis.

Covolumes of nonuniform lattices in PU( n , 1)

This paper studies the covolumes of nonuniform arithmetic lattices in PU( n , 1). We determine the smallest covolume nonuniform arithmetic lattices for each n , the number of minimal covolume lattices for each n , and study the growth of the minimal covolume as n varies. In particular, there is a unique lattice (up to isomorphism) in PU(9, 1) of smallest Euler-Poincaré characteristic amongst all nonuniform arithmetic lattices in PU( n , 1). We also show that for each even n there are arbitrarily large families of nonisomorphic maximal nonuniform lattices in PU( n , 1) of equal covolume.

Arithmeticity of complex hyperbolic triangle groups

Complex hyperbolic triangle groups, originally studied by Mostow in building the rst nonarithmetic lattices in PU(2 ; 1), are a natural generalization of the classical triangle groups. A theorem of Takeuchi states that there are only nitely many Fuchsian triangle groups that determine an arithmetic lattice in PSL2(R), so triangle groups are generically nonarithmetic. We prove similar niteness theorems for complex hyperbolic triangle groups that determine an arithmetic lattice in PU(2; 1).

Cone metrics on the sphere and Livné’s lattices

We give an explicit construction of a family of lattices in PU (1, 2) originally constructed by Livne. Following Thurston, we construct these lattices as the modular group of certain Euclidean cone metrics on the sphere. We give connections between these groups and other groups of complex hyperbolic isometries.

Complex kleinian groups and limit sets in P 2

For Γ a lattice in PU(2,1), the isometries of the 2-complex ball naturally extend to an action in : we prove that the limit set of this action coincides with the complement of the ball in the projective space. We prove that the action is ergodic with respect to a certain measure. Also, we study other kinds of actions: for instance we construct an action whose limit set is the entire , but is not minimal although it is algebraically mixing.

Virtual first Betti number and integrality of compact complex two-ball quotients

We prove that for a compact complex hyperbolic space form of complex dimension two whose associated lattice is nonintegral, the first Betti number is virtually positive. This provides support to a question of Borel in favor of a conjecture of Thurston on virtual positivity of the first Betti number on complex hyperbolic space forms. The result can also be considered as a topological sufficient condition for the integrality of a lattice in PU(2,1).

Sur la rigidit� de certains groupes fondamentaux, l?arithm�ticit� des r�seaux hyperboliques complexes, et les ?faux plans projectifs?

The motivation of this work comes from the study of lattices in real simple Lie groups. The famous Margulis’s superrigidity theorem claims that finite dimensional reductive representations of any lattice of a real simple Lie group of real rank ≥2 are superrigid. As a corollary such a lattice is arithmetic. These results extend to the real rank one case for lattices in Sp(n,1) and F4(-20) by the work of Corlette and Gromov-Schoen. On the other hand Mostow and Deligne-Mostow exhibited arithmetic lattices with non-superrigid representations as well as non-arithmetic lattices in the unitary group PU(2,1). A natural question is then to find simple sufficient conditions for superrigidity or arithmeticity of lattices in PU(2,1). Rogawski conjectured the following: let Γ be a torsion-free cocompact lattice in PU(2,1) such that the hyperbolic quotient M=Γ\B2ℂ verifies the cohomogical conditions b1(M)=0 and H1,1(M,ℂ)∩H2(M,ℚ)≃ℚ. Then Γ is arithmetic. In this paper we consider a smooth complex projective surface M verifying the above cohomological assumptions and study Zariski-dense representations of the fundamental group π1(M) in a simple k-group H of k-rank ≤2 (where k denotes a local field). Our main result states that there are strong restrictions on such representations, especially when k is non-archimedean (Theorem 5). We then consider some cocompact lattices in PU(2,1) of special geometric interest: recall that a “fake P2ℂ” is a smooth complex surface (distinct from P2ℂ) having the same Betti numbers as P2ℂ. “Fake P2ℂ” exist by a result of Mumford and are complex hyperbolic quotients Γ\H2ℂ by Yau’s proof of the Calabi conjecture. They obviously verify the hypotheses of Rogawski’s conjecture. In this case we prove that every Zariski-dense representation of Γ in PGL(3) is superrigid in the sense of Margulis (Theorem 3). As a corollary every “fake P2ℂ” is an arithmetic quotient of the ball B2ℂ.

Isomorphisms among monodromy groups and applications to lattices in PU(1,2)

Isomorphisms among monodromy groups and applications to lattices in PU(1<i>,</i>2)