Complex Hyperbolic Lattices Research Articles

Subgroups of hyperbolic groups, finiteness properties and complex hyperbolic lattices

We prove that in a cocompact complex hyperbolic arithmetic lattice Γ<PU(m,1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\Gamma < {\\mathrm{PU}}(m,1)$\\end{document} of the simplest type, deep enough finite index subgroups admit plenty of homomorphisms to ℤ with kernel of type Fm−1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathscr{F}_{m-1}$\\end{document} but not of type Fm\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathscr{F}_{m}$\\end{document}. This provides many finitely presented non-hyperbolic subgroups of hyperbolic groups and answers an old question of Brady. Our method also yields a proof of a special case of Singer’s conjecture for aspherical Kähler manifolds.

ARITHMETICITY OF -FUCHSIAN SUBGROUPS OF SOME NONARITHMETIC LATTICES

AbstractWe study the arithmeticity of $\mathbb {C}$ -Fuchsian subgroups of some nonarithmetic lattices constructed by Deraux et al. [‘New non-arithmetic complex hyperbolic lattices’, Invent. Math.203 (2016), 681–771]. Our results give an answer to a question raised by Wells [Hybrid Subgroups of Complex Hyperbolic Isometries, Doctoral thesis, Arizona State University, 2019].

Arithmeticity, superrigidity and totally geodesic submanifolds of complex hyperbolic manifolds

For $$n \ge 2$$ , we prove that a finite volume complex hyperbolic n-manifold containing infinitely many maximal properly immersed totally geodesic submanifolds of real dimension at least two is arithmetic, paralleling our previous work for real hyperbolic manifolds. As in the real hyperbolic case, our primary result is a superrigidity theorem for certain representations of complex hyperbolic lattices. The proof requires developing new general tools not needed in the real hyperbolic case. Our main results also have a number of other applications. For example, we prove nonexistence of certain maps between complex hyperbolic manifolds, which is related to a question of Siu, that certain hyperbolic 3-manifolds cannot be totally geodesic submanifolds of complex hyperbolic manifolds, and that arithmeticity of complex hyperbolic manifolds is detected purely by the topology of the underlying complex variety, which is related to a question of Margulis. Our results also provide some evidence for a conjecture of Klingler that is a broad generalization of the Zilber–Pink conjecture.

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.

Boundary maps and maximal representations on infinite-dimensional Hermitian symmetric spaces

AbstractWe define a Toledo number for actions of surface groups and complex hyperbolic lattices on infinite-dimensional Hermitian symmetric spaces, which allows us to define maximal representations. When the target is not of tube type, we show that there cannot be Zariski-dense maximal representations, and whenever the existence of a boundary map can be guaranteed, the representation preserves a finite-dimensional totally geodesic subspace on which the action is maximal. In the opposite direction, we construct examples of geometrically dense maximal representation in the infinite-dimensional Hermitian symmetric space of tube type and finite rank. Our approach is based on the study of boundary maps, which we are able to construct in low ranks or under some suitable Zariski density assumption, circumventing the lack of local compactness in the infinite-dimensional setting.

Multiplicative constants and maximal measurable cocycles in bounded cohomology

AbstractMultiplicative constants are a fundamental tool in the study of maximal representations. In this paper, we show how to extend such notion, and the associated framework, to measurable cocycles theory. As an application of this approach, we define and study the Cartan invariant for measurable$\mathrm{PU}(m,1)$-cocycles of complex hyperbolic lattices.

Superrigidity of maximal measurable cocycles of complex hyperbolic lattices

Let Gamma be a torsion-free lattice of PU (p,1) with p ge 2 and let (X,mu _X) be an ergodic standard Borel probability Gamma -space. We prove that any maximal Zariski dense measurable cocycle sigma : Gamma times X longrightarrow SU (m,n) is cohomologous to a cocycle associated to a representation of PU (p,1) into SU (m,n), with 1 le m le n. The proof follows the line of Zimmer’ Superrigidity Theorem and requires the existence of a boundary map, that we prove in a much more general setting. As a consequence of our result, there cannot exist maximal measurable cocycles with the above properties when 1< m < n.

Toledo invariant of lattices in SU(2,1) via symmetric square

In this paper, we address the issue of the quaternionic Toledo invariant to study the character variety of two-dimensional complex hyperbolic uniform lattices into SU(n, 2), n ≥ 4. We construct four distinct representations to prove that the character variety contains at least seven distinct components. We also show the existence of holomorphic horizontal lift to various period domains of SU(n, 2).

Maximal representations of cocompact complex hyperbolic lattices, a uniform approach

We complete the classification of maximal representations of cocompact complex hyperbolic lattices in Hermitian Lie groups by dealing with the exceptional groups E 6(-14) and E 7(-25) . We prove that if ρ is a maximal representation of a cocompact complex hyperbolic lattice Γ⊂SU(1,n), n>1, in an exceptional Hermitian group G ℝ , then n=2 and G ℝ =E 6(-14) , and we describe completely the representation ρ. The case of classical Hermitian target groups was treated by Vincent Koziarz and the second named author [KM17]. However we do not focus immediately on the exceptional cases and instead we provide a more unified perspective, as independent as possible of the classification of the simple Hermitian Lie groups. This relies on the study of the cominuscule representation of the complexification G of the target group G ℝ . As a by-product of our methods, when the target Hermitian group G ℝ has tube type, we obtain an inequality on the Toledo invariant of the representation ρ:Γ→G ℝ which is stronger than the Milnor-Wood inequality (thereby excluding maximal representations in such groups).

Eichler–Shimura isomorphism for complex hyperbolic lattices

We consider the cohomology group H-1(Gamma, p) of a discrete subgroup Gamma subset of G = SU(n, 1) and the symmetric tensor representation p on S-m(c(n+1)). We give an elementary proof of the Eichler-Shimura isomorphism that harmonic forms H-1(Gamma G/K, p) are (0, 1)-forms for the automorphic holomorphic bundle induced by the representation S-m(C-n) of K.

Local rigidity of complex hyperbolic lattices in semisimple Lie groups

AbstractWe show the local rigidity of complex hyperbolic lattices in classical Hermitian semisimple Lie groups, SU(p, q), Sp(2n + 2, $\mathbb{R}$), SO*(2n + 2), SO(2n, 2). This reproves or generalises some results in [2, 9, 11, 15].

Maximal representations of uniform complex hyperbolic lattices

Let $\rho$ be a maximal representation of a uniform lattice $\Gamma\subset{\rm SU}(n,1)$, $n\geq 2$, in a classical Lie group of Hermitian type $H$. We prove that necessarily $H={\rm SU}(p,q)$ with $p\geq qn$ and there exists a holomorphic or antiholomorphic $\rho$-equivariant map from complex hyperbolic space to the symmetric space associated to ${\rm SU}(p,q)$. This map is moreover a totally geodesic homothetic embedding. In particular, up to a representation in a compact subgroup of ${\rm SU}(p,q)$, the representation $\rho$ extends to a representation of ${\rm SU}(n,1)$ in ${\rm SU}(p,q)$.

Maximal representations of complex hyperbolic lattices into SU(m,n)

Let Γ denote a lattice in SU(1, p), with p greater than 1. We show that there exists no Zariski dense maximal representation with target SU(m, n) if n > m > 1. The proof is geometric and is based on the study of the rigidity properties of the geometry whose points are isotropic m-subspaces of a complex vector space V endowed with a Hermitian metric h of signature (m, n) and whose lines correspond to the 2m dimensional subspaces of V on which the restriction of h has signature (m, m).

New non-arithmetic complex hyperbolic lattices

We produce a family of new, non-arithmetic lattices in $${\mathrm{PU}}(2,1)$$ . All previously known examples were commensurable with lattices constructed by Picard, Mostow, and Deligne–Mostow, and fell into nine commensurability classes. Our groups produce five new distinct commensurability classes. Most of the techniques are completely general, and provide efficient geometric and computational tools for constructing fundamental domains for discrete groups acting on the complex hyperbolic plane.

Symmetric differentials, Kähler groups and ball quotients

While Margulis’ superrigidity theorem completely describes the finite dimensional linear representations of lattices of higher rank simple real Lie groups, almost nothing is known concerning the representation theory of complex hyperbolic lattices. The main result of this paper (Theorem 1.3) is a strong rigidity theorem for a certain class of cocompact arithmetic complex hyperbolic lattices. It relies on the following two ingredients: Theorem 1.6 showing that the representations of the topological fundamental group of a compact Kahler manifold X are controlled by the global symmetric differentials on X. An arithmetic vanishing theorem for global symmetric differentials on certain compact ball quotients using automorphic forms, in particular deep results of Clozel on base change (Theorem 1.11).

Invariant Symmetries of Unimodal Function Singularities

We classify finite order symmetries g of the 14 exceptional unimodal function singularities f in 3 variables, which satisfy a so-called splitting condition. This means that the rank 2 positive subspace in the vanishing homology of f should not be contained in one eigenspace of g?. We also obtain a description of the hyperbolic complex reflection groups appearing as equivariant monodromy groups acting on the hyperbolic eigensubspaces arising. One of the most famous classical results in singularity theory is the Arnold and Brieskorn discovery of the close relationship between simple function singularities and Weyl groups Aμ, Dμ, Eμ [1, 6]. A few years after it, Arnold extended the relationship to simple singularities with the Z2 reflection symmetry and Weyl groups Bμ, Cμ, F4 [2] (see also Slodowy’s book [24]). Consideration of Zm symmetries of simple functions led in [10, 11, 12, 25] to the appearance of Shephard-Todd groups within function singularities. The emphasis there was on realisations of the complex reflection groups as equivariant monodromy groups acting on the appropriate character subspaces in the homology of invariant Milnor fibres, and on the diffeomorphisms between the discriminants of the reflection groups and of the Zm-equivariant functions. A further series of papers [13, 14, 15], on cyclic symmetries of the parabolic functions, brought in similar singularity realisations of certain complex crystallographic groups [22]. In this paper, we are naturally expanding the programme to cyclic symmetries of the 14 exceptional unimodal function singularities on one hand, and complex hyperbolic reflection groups on the other. The basic idea is as follows. In the 3-variable case, the intersection form on the vanishing homology of an exceptional unimodal function f is non-degenerate and has positive signature 2. Assume g is an automorphism of C of finite order m, and our function is g-invariant. Then g acts on the second homology of the Milnor fibre f−1(e), and decomposes it into a direct sum of the character subspaces Hχ, χ = 1, on which g acts as multiplication by χ. Assume the rank 2 positive subspace of the intersection form splits between two character summands. Then the monodromy within a g-invariant versal deformation of f acts as a complex hyperbolic reflection group on each of them. Developing further the technique introduced in papers on cyclically symmetric functions [10, 11, 12], we construct vanishing bases in the hyperbolic summands and obtain the generating reflections as the corresponding Picard-Lefschetz operators. The main result of the paper is a complete classification of the invariant symmetries of the 14 singularities, which split the positive subspace in the vanishing homology, and the description – via constructing the corresponding Dynkin diagrams – of the complex hyperbolic groups arising. All the rank 2 reflection groups obtained projectivise to the triangle groups of the Poincare disk. The task of identification of higher dimensional groups is left for a future paper, along with the consideration of the equivariant symmetry setting. It should be noted that it is the first time when complex hyperbolic reflection groups are appearing in a singularity theory context. The approach introduced may be useful for constructing new complex hyperbolic lattices (cf. [8, 20]). The paper is organised as follows. Section 1 introduces the notion of singularities with symmetry, recalls the definitions and constructions given in [10, 11, 12]. Section 2 contains classification of splitting invariant symmetries of the 14 singularities. In Section 3.3 we construct Dynkin diagrams of the hyperbolic monodromy groups associated with the symmetric functions. Projectivisations of the rank 2 monodromy groups are considered in Section 4. More details of the constructions may be found in [16].

A Milnor–Wood inequality for complex hyperbolic lattices in quaternionic space

We prove a Milnor-Wood inequality for representations of the fundamental group of a compact complex hyperbolic manifold in the group of isometries of quaternionic hyperbolic space. Of special interest is the case of equality, and its application to rigidity. We show that equality can only be achieved for totally geodesic representations, thereby establishing a global rigidity theorem for totally geodesic representations.

Local rigidity for complex hyperbolic lattices and Hodge theory

We prove general local rigidity results for cocompact complex hyperbolic lattices using Hodge theory.

Local quaternionic rigidity for complex hyperbolic lattices

AbstractLet $\smash{\sGa\stackrel{i}{\hookrightarrow}L}$ be a lattice in the real simple Lie group L. If L is of rank at least 2 (respectively locally isomorphic to Sp(n, 1)) any unbounded morphism ρ : Γ → G into a simple real Lie group G essentially extends to a Lie morphism ρL : L → G (Margulis's superrigidity theorem, respectively Corlette's theorem). In particular any such morphism is infinitesimally, thus locally, rigid. On the other hand, for L = SU(n, 1) even morphisms of the form $\smash{\rho:\sGa\stackrel{i}{\hookrightarrow}L \rightarrow G}$ are not infinitesimally rigid in general. Almost nothing is known about their local rigidity. In this paper we prove that any cocompact lattice Γ in SU(n, 1) is essentially locally rigid (while in general not infinitesimally rigid) in the quaternionic groups Sp(n, 1), SU(2n, 2) or SO(4n, 4) (for the natural sequence of embeddings SU(n, 1) ⊂ Sp(n, 1) ⊂ SU(2n, 2) ⊂ SO(4n, 4)).

The Toledo invariant on smooth varieties of general type

We propose a definition of the Toledo invariant for representations of fundamental groups of smooth varieties of general type into semisimple Lie groups of Hermitian type. This definition allows to generalize the results known in the classical case of representations of complex hyperbolic lattices to this new setting: assuming that the rank of the target Lie group is not greater than two, we prove that the Toledo invariant satisfies a Milnor-Wood type inequality and we characterize the corresponding maximal representations.