Representations of Deligne-Mostow lattices into PGL(3, C)

We classify representations of a class of Deligne-Mostow lattices into PGL(3;C). In particular, we show local rigidity for the representations (of Deligne-Mostow lattices with 3-fold symmetry and of type one) where the generators we chose are of the same type as the generators of Deligne-Mostow lattices. We also show local rigidity without constraints on the type of generators for six of them and we show the existence of local deformations for a number of representations in three of them. We use formal computations in SAGE and Maple to obtain the results. The code files are available on GitHub.


Introduction
Lattices in semi-simple Lie groups have strong rigidity properties (see [Mor15] for a general introduction to lattices in semi-simple Lie groups and more details in Section 2).The Mostow-Prasad rigidity theorem states that, when ρ : Γ 1 → Γ 2 is an isomorphism between two lattices Γ 1 , Γ 2 contained in G 1 and G 2 (with G i being a connected simple Lie group with trivial center and non-isogenous to SL(2, R)), then ρ extends to an isomorphism between the Lie groups G 1 and G 2 .In higher rank, Margulis' superrigidity theorem implies that any homomorphism φ : Γ → G ′ of a lattice Γ ⊂ G (with G, G ′ simple, connected, with trivial centre, no compact factors and such that the rank of G is > 1) with Zariski dense image, extends to a homomorphism φ : G → G ′ .
In rank one, it is well known that superrigidity fails.In particular, there exist representations of a given lattice in PU(2, 1) into other Lie groups which do not extend to homomorphisms of the whole group PU(2, 1) (see Remark 2.5).In this paper we are interested in representations (up to conjugation) of a class of Deligne-Mostow lattices into PGL(3, C).
An important motivation for our results is the study of complex projective structures.Indeed, given a complex projective structure on a complex surface M , one can define a developing map D : M → CP 2 and a holonomy representation ρ : π 1 (M ) → PGL(3, C).Complex hyperbolic manifolds carry an induced complete complex projective structure.It is an open question whether a complex hyperbolic manifold has a different complex projective structure than this one.On the other hand, fixing the complex structure of a finite volume complex hyperbolic surface, its complex projective structure is unique by a theorem of Mok-Yeung [MY93] (see also [Kli01]).This means that if a complex hyperbolic manifold carried a complex projective structure different from the one induced by the complex hyperbolic structure, it would have to come from a different complex structure.It is possible that, for the non-cocompact lattices, some of the representations in our list correspond (taking torsion free subgroups) to complex projective structures on complex manifolds diffeomorphic but not biholomorphic to complex hyperbolic manifolds.One should be aware that in the compact case, Siu's rigidity theorem (valid for any compact hermitian locally symmetric space) implies that any other Kähler complex structure on the manifold is biholomorphic or conjugate-biholomorphic to the original complex hyperbolic structure.
A more comprehensive overview of the panorama of known rigidity theorems that motivate our work and relate to it is given in Section 2.
In section 3 we state the classification of all representations where the generators we chose are of the same type as the generators of the Deligne-Mostow lattice.
In Section 4 we review the definition of the Deligne-Mostow lattices with 3-fold symmetry in PU(2, 1).The main result which allows for our computations is the explicit presentation of the 3-fold Deligne-Mostow lattices.It was obtained in several steps culminating in [Pas16], where a unified treatment of all 3-fold Deligne-Mostow lattices is given.We will use a presentation of each Deligne-Mostow group using two generators easily obtained from the presentations in [Pas16].
In section 5 we give the detailed proofs for one case.The other cases are calculated in the same way and can be found in the SAGE and MAPLE companion notebooks to this paper.The main strategy is to use Gröbner basis methods to solve the equations imposed by the relators in the presentation.We use computations in SAGE and in MAPLE which are available on GitHub ( [FPUP21a]).The case of representations not preserving generator types is treated in the Maple files.In these, we make use of the Rational Univariate Representation to get formal parameterizations of the solutions when the system we obtain is 0-dimensional (for an introduction to the method in the context of representations to PGL(3, C) see [FKR15]).The computations show that the representations obtained for six of the lattices are all locally rigid (that is, 0-dimensional), while for three of them, there exists a 1-dimensional branch of representations.We were not able to find explicit parametrizations of them but they are given by explicit Gröbner basis in the available MAPLE file (see also the comments at the end of Section 3).With the exception of the Deligne-Mostow lattice itself (because of the local rigidity theorem for complex hyperbolic lattices -see Lemma 2.2), we do not know if this local rigidity phenomenon for type preserving representations is specific to the lattices we studied or has a more general scope.With an appropriate conjugation, the representations have values in an algebraic field extension of Q and we studied the orbits by the Galois group.
We thank Sorin Dumitrescu for useful discussions and for pointing to us the local rigidity result in [MY93].The second author also acknowledges the support of the FSMP and of the EPSRC grant EP/T015926/1.The third author acknowledges the support of the COnsejo NAcional de Ciencia Y Tecnología, project 769037.We also thank the anonymous referees for useful comments and remarks.

Rigidity of lattices and projective structures
In this section we give some background on the problem of finding representations of a lattice and several fundamental rigidity theorems which are known.
Representations of a group Γ into a Lie group G have been studied for a long time.It is common to consider only representations modulo conjugation by the Lie group and refer to the set of deformations modulo conjugation as the deformation variety.For instance, if Γ is the fundamental group of a compact surface and G = SL(2, R), then one component of the deformation variety is Teichmüller space, containing only discrete and faithful representations.On the other hand, assume that G is a higher dimensional semi-simple group and that Γ is a lattice in a Lie group (that is, a discrete subgroup with a finite volume quotient for the Haar measure).Then, building on partial results by Calabi, Selberg, Calabi-Vesentini, the following local rigidity result was obtained by A. Weil in the cocompact case and by Garland-Raghunatan in the non-cocompact case.
Theorem 2.1.Let ρ : Γ → G be an isomorphism onto its image, which is a lattice in the semi-simple Lie group G.If G is not locally isomorphic to SL(2, R) or SL(2, C) then the embedding is locally rigid.In other words, all representations near the embedding are conjugate.
A slightly different but strongly related rigidity concept is that of infinitesimal rigidity.The infinitesimal rigidity of a representation ρ : Γ → G is, by definition, the vanishing of the cohomology group H 1 (Γ, g), where Γ acts on the Lie algebra of G, g, by the adjoint representation Ad ρ : G → Aut(g).The cohomology group is identified with the Zariski tangent space of the deformation variety.Then Weil's criterion states the following: if Γ is finitely generated, then an infinitesimally rigid representation is also locally rigid.Note that here we are not assuming that G is semi-simple nor that Γ is a lattice.
Proof.By Weil's criterion it suffices to prove that the representation is infinitesimaly rigid, that is This lemma, with the same proof, is also valid for other representations of lattices obtained through complexifications of Lie groups.Related to local rigidity of complex hyperbolic lattices one should mention that the complex structure of a quotient Γ\H n C of complex hyperbolic space by a torsion free lattice is locally rigid [CV60].It is still an open problem to decide if this complex structure is unique.
A step further is the passage from local rigidity to global rigidity in the case of lattices.This is known as the Mostow theorem and is due to Mostow for the cocompact case and to Prasad for the non-cocompact case (see [Mor15] pg.309): Theorem 2.3.Let ρ : Γ 1 → Γ 2 be an isomorphism between two lattices in G 1 and G 2 (with G i connected non-compact simple Lie group with trivial center non-isogenous to SL(2, R)).Then ρ extends to an isomorphism between the Lie groups G 1 and G 2 .
In particular, in the case of two lattices Γ 1 and Γ 2 in PU(n, 1), Γ 2 is conjugated to Γ 1 or to its complex conjugate by an element of PU(n, 1).
The main goal of this paper is to classify representations of lattices in PU(2, 1) into PGL(3, C).By Mostow-Prasad rigidity, they are rigid in PU(2, 1).But, as it turns out, they are not rigid in PGL(3, C).This should be compared with the superrigidity theorem by Margulis.We state it in a simplified form (see [Mor15] pg.323): Theorem 2.4.Assume G 1 and G 2 are connected non-compact simple Lie groups with trivial centre.Assume that the rank of The theorem is also valid if G 1 is a Lie group of rank one other than those isogeneous to SU(n, 1) or SO(n, 1) (see [Cor92]).The techniques used in the rank one case are inspired by Siu's theorem [Siu80] which states that any Kähler manifold homotopic to a compact complex hyperbolic manifold is either biholomorphic or conjugate biholomorphic to it.
Remark 2.5.Mostow (see [Mos80] section 22) constructed two cocompact lattices (which are arithmetic) Γ 1 and Γ 2 of PU(2, 1) and a surjective homomorphism ρ : Γ 1 → Γ 2 with infinite kernel.This gives an example of a violation of superrigidity for lattices in PU(2, 1).We are not able to exactly relate this homomorphism with our representations, since our group is the same as Mostow's group only up to finite index.

Real and complex projective structures.
In this paper we construct a large class of homomorphisms of Deligne-Mostow lattices in PU(2, 1) into PGL(3, C).One might think that these representations (taking torsion free subgroups of the lattice) might be holonomy representations of different complex projective structures on the quotient of complex hyperbolic space by the lattice.But if the complex structure of the quotient is fixed, that cannot be the case: C be a complex hyperbolic manifold of finite volume.Then its complex projective structure induced by the embedding H 2 C ⊂ CP 2 is the unique projective structure compatible with its complex structure.
On the other hand, the existence of different complex structures on the quotient Γ\H 2 C (for a torsion free lattice Γ) is an open problem.The theorem is also true for complex hyperbolic manifolds of dimension greater than two (see [MY93]) and for higher rank hermitian symmetric domains (see [Kli01]).
It is interesting to compare the analogous situation arising in the case of lattices in PO(n, 1).One can consider the embedding PO(n, 1) ⊂ PGL(n + 1, R) and look for deformations.The argument in Lemma 2.2 does not apply anymore and indeed in many situations the lattice is not locally rigid in PGL(n + 1, R) (see [JM87]).In particular, the existence of totally geodesic hypersurfaces in a quotient of real hyperbolic space, Γ\H n R implies the existence of deformations, called bendings, of the group Γ inside PGL(n + 1, R).The space of deformations of representations of Γ into PGL(n + 1, R) modulo conjugation has been studied for a long time with contributions by several authors (see the survey [Qui10] which describes, among others, contributions by Kac-Vinberg, Koszul and Benoist).In particular, starting from a cocompact discrete subgroup of PO(n, 1), there exists a whole component of the space of deformations consisting of discrete faithful representations ρ of Γ such that there exists a convex open subset in projective space Ω ⊂ RP n whose quotient by ρ(Γ) is compact.These deformations correspond to holonomies of real projective structures on the underlying manifold structure of the hyperbolic structure.

Other deformations.
Another problem related to the one we study is to classify embeddings of lattices in PU(n, 1) into PU(n + 1, 1) or the analogous problem of embeddings of lattices of PO(n, 1) into PO(n + 1, 1).For the latter, bending deformations were also studied in [JM87].The former has been studied in [GM87] and [Tol89].In particular, a discrete, torsion free, cocompact subgroup Γ ⊂ PU(n, 1) embedded in PU(n + 1, 1) preserving a totally geodesic complex hypersurface in complex hyperbolic space has some rigidity in the sense that a connected component of the representation space into PU(n + 1, 1) consists also of discrete representations preserving a totally geodesic complex hypersurface.For a review and generalizations of this phenomenon we refer to [KM17].See also [Poz15] for a recent overview of maximal representations of lattices in PU(1, 1) into PU(2, 1).

The results
We refer to section 4 for the definition of Deligne-Mostow lattices.For the statement of the results we simply recall that 3-fold Deligne-Mostow lattices of type one are parametrised by a pair of integers (p, k) and are generated by two elements J and R 1 which are, respectively, a regular elliptic element of order 3 and a complex reflection of order p.We say that a representation of the lattice preserves generators types if the image of a complex reflection generator (respectively regular elliptic) is a complex reflection (respectively regular elliptic).Note that in this definition the image of a complex reflection could be the identity and its order could be a a factor of the order of the Deligne-Mostow generator.That is, conjugacy classes of reflections might not be preserved.1. Representations preserving generator types are locally rigid and are classified in Tables 1 and 2.
2. Representations without constraints on generators are also locally rigid, except for lattices (4,3), (4,4) and (6,2) which have one dimensional branches for certain representations with regular elliptic generators.They are enumerated in the MAPLE file in [FPUP21a].
In Table 1 are the details for each compact 3-fold type one Deligne-Mostow lattice.The representations are defined with coefficients in a cyclotomic extension of Q, which depends on (p, k).We also compute the orbits of the Galois group action on the space of representations.For the compact Deligne-Mostow lattices, there exists an invariant Hermitian form which is always non-degenerate.
(p, k) Total Galois Orbits Table 1: Compact 3-fold Deligne-Mostow Lattices In Table 2 are the details on non-compact 3-fold type one lattices.Each pair (p, k) defines a Q-extension and each representation of the (p, k) lattice has coefficient in (a subfield of) this Q−extension.For each representation there exists an invariant Hermitian form that can be non-degenerate with signatures (2,1) or (3,0), or degenerate.While the irreducible representations can only come from non-degenerate configurations, the reducible ones can come from either degenerate or non-degenerate configurations (see Section 5).In the Reducible column the numbers (m, n) mean that m reducible representations come from non-degenerate configurations and n come from degenerate ones.The last column corresponds to the number of representations which contain in their kernel an element of the centraliser of the cusp group.One may interpret those representations as representations which factor through the orbifold fundamental group of a compactification of the complex hyperbolic orbifold defined by the Deligne-Mostow lattice.
Concerning the representations which do not preserve types of generators, the computations show that they are all locally rigid except for one dimensional branches for Deligne-Mostow lattices (4, 3), (4, 4) and (6, 2).We did not obtain, though, a parametrisation of the few examples having a positive dimension.In fact, we obtain the Hilbert dimension of the system of equations describing representations for each fixed order of the eigenvalues of R 1 (note that, for each Deligne-Mostow group of of type (p, k) the type preserving representations, R 1 has eigenvalues 1 and order p).In a few cases, described in detail in the MAPLE file, the Hilbert dimension is one.This implies that there exists a one dimensional branch (p, k) Total Galois Orbits
We also determine which representations may be lifted to GL(3, C) representations (see [FPUP21a]).The representations with generators ρ(J) and ρ(R 1 ) of other types are not tabulated.
In the following sections, we give details for the calculations in the case of the Deligne-Mostow lattice (3, 6).We also compute the reducible representations of this lattice when the generators are not of the same type as the ones of the Deligne-Mostow lattice (note that in this case, all representations are locally rigid).The same calculations for the other Deligne-Mostow lattices in the table can be found on GitHub ([FPUP21a]).

Deligne-Mostow lattices
The Deligne-Mostow lattices were introduced and studied by Mostow and by Deligne and Mostow in various works, including [Mos80] and [DM86].They have a long history dating back to by Picard, Le Vavasseur and Lauricella.Deligne and Mostow start with a ball N-tuple µ = (µ 1 , . . ., µ N ) which is a set of N real numbers in (0, 1) whose sum is 2. Then they look at a hypergeometric function defined using these parameters.Finally, they use the monodromy action to build some lattices in P U (N − 3, 1).This list of lattices contains the first known examples of non-arithmetic complex hyperbolic lattices in dimensions 2 and 3.
Here we will concentrate on a special class of the Deligne-Mostow lattices in PU(2, 1), called the lattices with 3-fold symmetry (or the 3-fold Deligne-Mostow lattices) and of type one.More precisely, first we restrict to the case of dimension 2. This means that we are looking at the group PU(2, 1) acting on 2-dimensional complex hyperbolic space and hence that we are looking at ball 5-tuples.In this dimension, the finite list of lattices that Deligne and Mostow found contains only lattices with some special symmetry.These are called the 2-fold and 3-fold symmetry lattices.A lattice has m-fold symmetry if m of the N elements of the ball N -tuple are equal.
In this work we will only look at some of the lattices with 3-fold symmetry.For all of those, one can find a construction for a fundamental domain and a presentation in [Pas16].In principle the 2-dimensional Deligne-Mostow lattices depend on the 5 elements of the ball 5-tuple.The condition on the sum of the elements of the ball 5-tuple reduces the parameters to 4. Of these 4, 3 are equal, so the lattices only depend on 2 parameters.One common choice for the parameters is to choose the orders of some of the generators.We will hence identify the lattices using a pair (p, k) which corresponds to the ball 5-tuple The Deligne-Mostow lattices in PU(2, 1) with 3-fold symmetry are divided in 4 types according to the ranges of p and k.The type determines the presentation, the volume formula and the combinatorics of a fundamental domain.Here we will look at the lattices of type one, for which an explicit fundamental domain and presentation can be found, for example, in [BP15].We are choosing these because they have minimal complexity (minimum number of facets in a fundamental domain, shortest presentation).Note that they are all arithmetic.In terms of p and k they are characterised by Then a presentation for them is given by We can rewrite this presentation in terms of the two generators J and R 1 in the following way: One can go back to the previous presentation writing R 2 = JR 1 J −1 and P = R 1 R 2 .
For the Deligne-Mostow lattices, the generator J is always regular elliptic of order 3, while the generator R 1 is a complex reflection of order p. Recall that by the classification of isometries in complex hyperbolic space (see, for example, [CG74]), elliptic elements have at least one fixed point inside H 2 C , parabolic elements have a unique fixed point on the boundary and loxodromic elements have exactly two fixed points on the boundary.Elliptic elements have finite order and have all eigenvalues of norm 1.They can be of two types: regular elliptic when they have three distinct eigenvalues, or complex reflections when they have a repeated eigenvalue.
The same analysis can be carried out for the Deligne-Mostow lattices of the other three types.This can be found in the note [FPUP21b].
5 The group (3,6) In this section we look in detail at the case (p, k) = (3, 6).The presentation for this case is: We first note that for the Deligne-Mostow lattices, one has that J is a regular elliptic element, while R 1 is a complex reflection.In general, while the presentation guarantees that ρ(J) and ρ(R 1 ) are both elliptic, we cannot say more about their type.We will hence look at all possible combinations of types: • when ρ(J) is regular elliptic and ρ(R 1 ) is a complex reflection, • when ρ(J) is a complex reflection and ρ(R 1 ) is regular elliptic, • when ρ(J) and ρ(R 1 ) are both regular elliptic, • when ρ(J) and ρ(R 1 ) are both complex reflections.
Note that if ρ(J) and ρ(R 1 ) are both complex reflections, then the group preserves the intersection of two complex lines in CP 2 .The group is therefore reducible and we will hence ignore this last case.5.0.1 ρ(J) is regular elliptic and ρ(R 1 ) is a complex reflection First, let us assume that the fixed points of ρ(J) do not intersect the fixed points of ρ(R 1 ).We will call these the non-degenerate configurations.We remark that this case might give rise to reducible representations when there exists a degenerate Hermitian form preserved by the generators with non-trivial kernel.We note in the following ω = − 1 2 + √ 3 2 i and ζ 9 a primitive 9th-root of unity.
Proposition 5.1.A representation ρ : Γ → PGL(3, C) such that ρ(R 1 ) is a complex reflection and ρ(J) has three distinct eigenvectors which do not intersect the fixed line of ρ(R 1 ) are given, up to conjugation, by the matrices Remark 5.2.
• The general procedure in the SAGE notebooks goes as follows.With the group relations we produce a system of equations and using SAGE we construct a polynomial ideal over Q[r 1 , r 2 , x].Then we produce a Gröebner basis for our system of equations.This has, as output, an array of polynomials that generates our initial ideal.Finally, we solve these polynomials for each variable to find the solutions.
• For each solution α j , we take lifts into SL(3, C) of the form: In order to have a lift of the representation, the previous matrices should satisfy the group relations.Therefore, we use the relations to obtain a system of equations in the variables k and l.It is interesting to note that in the (3,6) case only the first six representations can be lifted to GL(3, C) representations (see [FPUP21a]).
We now look at the degenerate configurations, where one eigenvector of ρ(J) intersects the fixed complex line of ρ(R 1 ).This gives rise to reducible representations, but note that they might not be completely reducible.
Proof.We choose a diagonal form for ρ(J) and impose that ρ(R 1 ) has the same eigenvector e 1 .One can then impose that the fixed line in projective space passes through [1, 0, 0] and [1, 1, 1].By conjugating by a diagonal matrix, one can suppose that ρ(R 1 ) is as above with r 2 to be determined.As in the previous proposition, a computation using Gröebner basis package in SAGE gives the result.Note that the two sets of solutions are complex conjugates.
Remark 5.4.Note that for solutions with the second form of R 1 in (8), the image of the group J, R 1 is contained in a copy of SL(2, C) inside PSL(3, C). implies that, for each solution, the group is contained in a PSL(2, C)−representation of the triangle group (3, 3, 12).In general, for the (p, k)-lattice, we can say that the image is contained in a PSL(2, C)−representation of a triangle group (3, p, 2k).
Note that if we remove the condition x 2 +x+1 = 0 in the previous proposition, there exists a unique (extra) solution for which R 1 = Id.This solution is part of the representations with type-preserving generators, and whose image is a cyclic group of order three generated by J.For the lattice (3,6), this is the unique reducible representation with finite image.This unique representation is contained also in representations for lattices (4,3), (5,3), (6,2), and (6,3).This solution is already taken into account in Tables 1 and 2. 5.0.2 ρ(J) is a complex reflection and ρ(R 1 ) is regular elliptic Interchanging the type of ρ(J) and ρ(R 1 ), a computation with Gröbner basis proves the following: Proposition 5.5.There are no representations ρ : Γ → PGL(3, C) such that ρ(J) is a complex reflection and ρ(R 1 ) has three distinct eigenvectors which do not intersect the fixed line of ρ(R 1 ).5.0.3 ρ(J) and ρ(R 1 ) are regular elliptic The final case to consider occurs when both ρ(J) and ρ(R 1 ) are regular elliptic.A simple computation shows that the two regular elliptic elements cannot have the same eigenspaces.
Proposition 5.6.The representations ρ : Γ → PGL(3, C) such that ρ(R 1 ) and ρ(J) are regular elliptic with at least one distinct eigenspace are given, up to conjugation, by the matrices and The cusp holonomy might contain elliptic elements.The purely parabolic cusp holonomy is the maximal subgroup of the cusp holonomy with no elliptics.
A straightforward computation with the list of representations gives the following description of cusp groups.The representations for which the generator of the centraliser is elliptic, may be factorised (up to a finite index subgroup) through a representation of the fundamental group of the Satake-Baily-Borel compactification.Indeed, in the compactification, the centraliser of the cusp holonomy disappears.
Proposition 5.10.The generator of the centraliser, (R 2 A 1 ) 2 , is elliptic of order at most three for all representations except for three (up to conjugation) where it can be chosen to be unipotent of the form (11)