Residually Finite Lattices in $\Widetilde {\Mathrm {PU}(2, 1)} $ And

$Residually Finite Lattices in $\Widetilde {\Mathrm {PU}(2, 1)} $ And$

Residually ﬁnite lattices in PU(2^, 1) and fundamental groups of smooth projective surfaces

Matthew Stover Domingo Toledo Temple University University of Utah [email protected] [email protected]

May 28, 2021

To Gopal Prasad in celebration of his 75th birthday

Abstract 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).

1 Introduction

The purpose of this paper is to study 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. This follows a general theme used by the second arXiv:2105.12772v1 [math.AG] 26 May 2021 author [46], and in a different way by Nori (unpublished) and Catanese– Kollár[1], to build smooth projective varieties with fundamental group that is not residually finite. 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).

1 Let G be an adjoint simple Lie group of hermitian type. Then Z ≤ π1(G), and the universal cover Ge of G ﬁts in a central extension

1 −→ Z −→ Ge −→ G −→ 1 for a certain finite cover G of G. Adjoint simple groups are linear, and it follows from a well-known theorem of Malcev [29] that any lattice Γ < G is residually finite. However, Ge is not linear and so it is not clear whether or not the preimage Γe of Γ in Ge is residually finite. When Γ is an arithmetic subgroup of an absolutely simple, simply connected algebraic group G defined over a number field K and Γ has the congruence subgroup property, Deligne [12] showed that the preimage Γe of Γ in Ge = Ge (R) need not be residually finite. If z is a generator for Z ≤ π1(G(R)) < Ge and n is the n order of the absolute metaplectic kernel M(∅, G), Deligne proved that z maps trivially to every finite quotient of Γ.e See work of Prasad and Rap- inchuk [33, p. 92] for a precise description of the connection between residual finiteness of Γe and | M(∅, G)| and for a complete description of M(∅, G). One then sees that the preimage of Γ in any connected d-fold cover of G(R) with d > n also cannot be residually finite. There are many known examples of lattices in PU(n, 1) without the congruence subgroup property, and it is widely expected to fail for all such lattices. Consequently, the following question seems interesting in contrast with the situation in higher-rank.

Question 1. Let G = PU(n, 1), Ge be its universal cover, Γ < G be a lattice, and Γe be the preimage of Γ in Ge. Is Γe residually ﬁnite? Is Γe linear?

We brieﬂy recall that Γe also has a natural geometric interpretation when n Γ is torsion-free and cocompact. Let B denote complex hyperbolic n-space. Then Γe is the fundamental group of the complement of the zero section n inside the total space of the canonical bundle of Γ\B . It can equivalently be considered as the fundamental group of the space

Γb\ SU(n, 1)/ SU(n), where Γb is the preimage of Γ in SU(n, 1) [27, §8.9]. See §3.4 for more on the connection with SU(n, 1)/ SU(n). We also note another motivation for Question 1 from geometric group theory. If there is a cocompact lattice Γe in Ge that is not residually finite, then, as in Deligne’s work [12], one could find a lattice in a finite cover of

2 PU(n, 1) that is not residually finite. This would be a finite central extension of a cocompact lattice in PU(n, 1), hence it would be a Gromov hyperbolic group that is not residually finite. Whether or not Gromov hyperbolic groups are residually finite is a long-standing open problem. It is known in the case n = 1 that Question 1 has a positive solution. ∼ In other words, lattices in the universal cover of PU(1, 1) = PSL2(R) are both residually finite and linear. Very briefly, the most well-known proof goes as follows. First, passing to a subgroup of finite index, we can assume that Γe is the fundamental group of the unit tangent bundle of a hyperbolic surface. Either geometrically or using a presentation for Γ,e one can produce a homomorphism from Γe to a two-step nilpotent group (in fact, an integer Heisenberg group) for which the restriction to the center of Γe is injective. Since Γ is residually finite, this suffices to conclude that Γe is also residually finite. See [11, §IV.48] for precise details from both the geometric and algebraic perspectives, and §2 for more on why finding such a quotient suffices to prove residual finiteness. The first aim of this paper is to construct the first examples of lattices in PU(2, 1) whose lift to the universal cover is residually finite. We in fact develop two methods for proving residual finiteness, each generalizing a variant of the classical argument for the case n = 1. Both strategies eventually intersect in the endgame of finding a two-step nilpotent quotient of Γe for which the central element survives to have infinite order. For torsion-free cocompact lattices, the two methods end up being effectively equivalent, but one or the other can be used more easily in different settings. We will prove:

Theorem 1.1. Let G = PU(2, 1) and Ge be its universal cover. There are lattices Γ < G so that the preimage of Γ in Ge is residually finite. Moreover, the preimage of Γ in any connected cover of G is residually finite. One can take Γ to be uniform or nonuniform. For a uniform arithmetic lattice, one can take the fundamental group of the Cartwright–Steger surface [7]. We prove this indirectly, using a 21-fold cover studied by the first author [41] and further explored by Dzambic and Roulleau [16]. See Theorem 5.8. A nonuniform arithmetic example is the Picard modular group over the Eisenstein integers. All of our examples are commensurable with Deligne–Mostow lattices [13, 32]. See §8. The first method of constructing examples, presented in §4, is more computational in nature. Replacing Γ with its preimage in SU(n, 1) and using the fact that Γ\ SU(n, 1)/ SU(n) has fundamental group Γ,e we describe a method for constructing a finite presentation for Γe using one for Γ. This is used in particular to provide the nonuniform examples in Theorem 1.1. Our

3 second method, inspired by ideas going back at least to work of Sullivan in ∗ n the 70s (see [2, Ch. 3]), uses the cup product structure on H (Γ\B , Q). The following, a special case of Corollary 5.4 and a consequence of a more general theorem about principal U(1) bundles over aspherical manifolds proved in §5, gives a sharp answer as to when the classical strategy for n = 1 can be used to prove residual finiteness of Γ.e Theorem 1.2. Let Γ < PU(n, 1) be a torsion-free cocompact lattice and n X = Γ\B . If Γe is the preimage of Γ in the universal cover of PU(n, 1), then there is a two-step nilpotent quotient of Γe that is injective on the center of Γe if and only if the Kählerclass on X is in the image of the cup product map ^2 1 2 cQ : H (X, Q) −→ H (X, Q). If Γ < PU(n, 1) is an arithmetic lattice defined by a hermitian form over a CM field, it is known that there is a torsion-free subgroup of finite index 0 0 0 n 1 0 Γ ≤ Γ so that X = Γ \B has H (X , Q) nontrivial [25, 37]. In fact, Clozel proved that one can pass to a possible smaller subgroup and assume that V2 1 0 2 0 the image of cQ : H (X , Q) → H (X , Q) is nontrivial [9]. It seems more subtle to ensure that the Kählerclass is eventually in the image. A positive answer to the following question would, via Theorem 1.2, prove residual finiteness of preimages of these arithmetic lattices in the universal cover of PU(n, 1). Question 2. Let Γ < PU(n, 1) be a cocompact arithmetic lattice defined by a hermitian form over a CM field. Is there a torsion-free finite index 0 0 0 n subgroup Γ ≤ Γ so that X = Γ \B has the property that the Kählerclass 0 V2 1 0 2 0 on X is in the image of the map cQ : H (X , Q) → H (X , Q)? The second part of this paper turns to applications to our understanding of fundamental groups of smooth projective surfaces. It is a common mis- conception that Siu’s rigidity theorem [38, Thm. 8] implies that any minimal smooth projective surface Y with the same fundamental group as a smooth compact ball quotient X must be biholomorphically or conjugate biholomorphically equivalent to X. In §6, we give the complete argument that this is the case under the additional hypothesis that π2(Y ) is trivial; this is perhaps well-known to experts, but we do not know a reference for this argument in the literature. In §7, we construct smooth projective surfaces whose fundamental groups fit into a central exact sequence

1 −→ Z/d −→ Γd −→ Γ −→ 1

4 for d ≥ 2 and Γ < PU(2, 1) a torsion-free cocompact lattice. More precisely, Γd is a lattice in the connected d-fold covering of PU(2, 1).

Theorem 1.3. Let Γ be a torsion-free cocompact lattice in PU(2, 1), Gd be its connected d-fold cover, and Γd be the preimage of Γ in Gd. Then Γd is the fundamental group of a smooth projective surface of general type that is not birational to a smooth compact ball quotient.

That Γd is not the fundamental group of a smooth compact ball quotient is clear from the fact that Γd contains elements of finite order. The construction is a variant of the one used by Nori (unpublished) and Catanese–Kollár [1]; see [2, Ex. 8.15]. We give full details of the construction for completeness. We prove Theorem 1.3 in §7, where it follows from the more general Theorem 7.4. When Γe is residually finite, passing to particular étale covers of the surfaces in Theorem 1.3 provides examples of smooth projective surfaces with the same fundamental group as a smooth compact ball quotient that are not birationally equivalent to one. One can also exploit the preimage of Γ in SU(2, 1), which is residually finite by linearity of SU(2, 1), to produce examples. Recent work of Troncoso and Urzúa[47] constructs many more smooth projective surfaces with fundamental group the same as a ball quotient. A key distinction between our work and [47] is that we build surfaces whose fundamental group is a lattice in a finite cover of PU(2, 1), which their work does not do, but [47] construct surfaces with the same fundamental group as a given ball quotient with Chern slopes dense in [1, 3], which is much more robust than what our methods produce (e.g., see Remark 7.6). In contrast, a torsion-free nonuniform lattice in PU(2, 1) cannot be the fundamental group of a smooth projective surface [45, Thm. 2]. As one specific application of our methods, we will prove the following at the conclusion of §7.

Theorem 1.4. Let X be a fake projective plane whose canonical divisor KX is divisible by three. Then there exists a minimal smooth projective surface ∼ Y of general type with π1(Y ) = π1(X) that is not a fake projective plane. Recall that fake projective planes were classiﬁed in combined work of Prasad–Yeung [34, 35] and Cartwright–Steger [7]. Of the 100 fake projective planes, all but 8 have the property that KX is divisible by three [35, §A.1]. However, we were not able to answer the following special case of Question 1.

Question 3. Let X be a fake projective plane and Γe be the preimage of π1(X) in the universal cover of PU(2, 1). Is Γe residually ﬁnite?

5 This paper is organized as follows. In §2 we begin with some basic facts about residual ﬁniteness. Then §3 describes explicit coordinates on the universal cover of SU(n, 1) that are used in §4 to describe our method for lifting a presentation of a lattice in SU(n, 1) to a presentation of its lift to the universal cover. The proof of Theorem 1.2 and generalizations are contained in §5. In §6, we prove that a smooth projective surface with π2 trivial and fundamental group a lattice in PU(2, 1) is biholomorphic to a ball quotient, and §7 proves Theorem 1.3 and related results. Finally, in §8 we discuss examples that suﬃce to prove Theorem 1.1.

Acknowledgments The authors thank Gopal Prasad for helpful correspondence regarding the metaplectic kernel. Stover was partially supported by Grant Number DMS- 1906088 from the National Science Foundation.

2 Residual ﬁniteness

Here, we collect some basic facts about residual finiteness. Recall that a group Γ is residually finite if, given any nontrivial γ ∈ Γ, there exists a finite group F and a homomorphism ρ :Γ → F so that ρ(γ) is not the identity element of F . Finitely generated linear groups are residually finite by a famous theorem of Malcev [29] (also see [28, §2.1]). We begin with a basic and well-known fact about residual finiteness; see for instance [28, Lem. 2.1.3] for a proof.

Lemma 2.1. Suppose that Γ is a finitely generated group and ∆ ≤ Γ is a finite index subgroup. Then Γ is residually finite if and only if ∆ is residually finite.

We will also need the following simple lemma.

Lemma 2.2. For n ≥ 2, suppose that

πn 1 −→ Z/n −→ Γn −→ Γ −→ 1 is a central exact sequence. If Γn is residually finite, then there exists a finite 0 index subgroup Γ ≤ Γ that admits a section to Γn under πn. Consequently, Γn has a finite index subgroup that is isomorphic to a finite index subgroup of Γ.

6 Proof. Suppose that Γn is residually ﬁnite and that σ is a generator for ker(πn). For each 1 ≤ j ≤ n − 1 there exists a ﬁnite group Fj and a j homomorphism ρj :Γn → Fj so that ρ(σ ) is nontrivial. The product homomorphism

n−1 Y ρ = (ρ1 × · · · × ρn−1):Γn −→ Fj j=1 then has the property that ρ(σj) is nontrivial for each 1 ≤ j ≤ n − 1. If 0 0 Γ = ker(ρ), then Γ ∩ hσi = {1}. Since hσi = ker(πn), we have that πn|Γ0 is an isomorphism onto its image. This proves the lemma.

Corollary 2.3. For any n ≥ 1, let Γ < SU(n, 1) be a finitely generated subgroup. Then there is a finite index subgroup Γ0 ≤ Γ so that projection SU(n, 1) → PU(n, 1) restricts to an isomorphism on Γ0. In particular, every lattice in SU(n, 1) contains a finite index subgroup isomorphic to a lattice in PU(n, 1).

Proof. Since SU(n, 1) is linear, Γ is residually ﬁnite. Let σ be a generator for the intersection of Γ with the center of SU(n, 1). The center of SU(n, 1) is isomorphic to Z/(n + 1), so hσi is cyclic. Therefore if Γ denotes the image of Γ in PU(n, 1), we have a central exact sequence

1 −→ hσi −→ Γ −→ Γ −→ 1 to which Lemma 2.2 applies, giving the desired Γ0.

Lemma 2.4. Suppose that Γ is a residually ﬁnite group and that

1 −→ Z −→ Γe −→ Γ −→ 1 is an extension. If there is a homomorphism ρ : Γe → H onto a residually ﬁnite group H so that ρ|Z is injective, then Γe is residually ﬁnite.

Proof. Given a nontrivial element α ∈ Γ,e if the projection α of α to Γ is nontrivial, then there is a map from Γ to a finite group where the image of α is nontrivial, hence there is a homomorphism from Γe to a finite group so that the image of α is nontrivial. If α is trivial, then α ∈ Z. Then ρ(α) ∈ H is nontrivial, and there is a finite quotient of H, which we consider as a finite quotient of Γ,e where the image of α is nontrivial. This proves that Γe is residually finite.

7 Finitely generated nilpotent groups are known to be residually finite. By Lemma 2.1, this follows from the fact that finitely generated nilpotent groups contain a torsion-free subgroup of finite index [22, Thm. 3.21] combined with Malcev’s famous results that finitely generated torsion-free nilpotent groups are linear [30] and that linear groups are residually finite [29]. The following is then a special case of Lemma 2.4.

Corollary 2.5. Suppose that Γ, Γe, and Z are as in Lemma 2.4. If Γe is ﬁnitely generated and there is a nilpotent quotient N of Γe so that Z injects into N , then Γe is residually ﬁnite.

We return to the case of interest for this paper, namely where Γe is a central extension of a finitely generated residually finite group Γ by Z. Corollary 2.5 implies that to prove Γe is residually finite, it suffices to find a nilpotent quotient of Γe into which Z injects. We also want to study the central exact sequence obtained by reducing the central element of Γe modulo an integer d. It is not immediately clear that residual finiteness of Γe implies residual finiteness of each Z/d central extension, only infinitely many of the extensions. Using separability properties of finitely generated nilpotent groups, we can prove the stronger statement:

Lemma 2.6. Suppose that Γ is a ﬁnitely generated residually ﬁnite group and that 1 −→ Z −→ Γe −→ Γ −→ 1 2 is a central exact sequence associated with φ ∈ H (Γ, Z). For d ∈ N, let 2 φd ∈ H (Γ, Z/d) be the reduction of φ modulo d and

1 −→ Z/d −→ Γd −→ Γ −→ 1 be the associated central exact sequence given by reducing the kernel hσi of Γe → Γ modulo d. Suppose that there is a nilpotent quotient N of Γe that is injective on hσi. Then Γd is residually finite for all d ∈ N. Proof. Let σ be a generator for the kernel of Γe → Γ and ρ :Γ → N be a ∼ nilpotent quotient such that S = hρ(σ)i = Z. Since σ is central in Γ,e ρ(σ) d is central in N . In particular, Sd = hρ(σ )i is a normal subgroup of N for every natural number d. Malcev proved that finitely generated nilpotent groups are subgroup separable [31], i.e., for every subgroup H ≤ N , there is a collection {Ni} of T finite index subgroups of N so that H = Ni. See [28, §2.3] for more basic facts about subgroup separability. Note that generally a group is subgroup

8 separable if this condition holds for all finitely generated subgroups H, but finitely generated nilpotent groups are Noetherian, hence all subgroups are finitely generated. By [36, Lem. 4.6], for each d ∈ N we can find a finite index subgroup Nd of N so that Nd ∩ S = Sd. Since Sd is normal in N , we can replace Nd by the intersection of all its conjugates in N and assume that Nd is normal in N . Then Fd = N /Nd is a finite quotient of Γe such that the image of hσi ∼ in Fd is isomorphic to S/(S ∩ Nd) = Z/d. This means that Γe → Fd factors through the projection Γe → Γd, and moreover that Γd → Fd is injective on the central Z/d quotient of hσi. Corollary 2.5 then implies that Γd is residually finite, which completes the proof of the lemma.

3 Structure of SU(n, 1) and its universal cover

This section gives precise coordinates, including an Iwasawa decomposition, for SU(n, 1) and its universal cover. These coordinates will be used to describe a concrete procedure for lifting ﬁnitely presented subgroups of SU(n, 1) to SU(^n, 1). Lastly, we study a certain homogeneous space for each group that will be fundamental for this lifting procedure. We start with some preliminaries on the universal cover U(]n) of the unitary group U(n).

3.1 Coordinates on U(gn)

The universal cover U(]n) of U(n) is identiﬁed with SU(n) × R under the universal covering map (g, t) 7→ eπitg. Composing this with the isomorphism g 7→ (g, det(g)−1) between U(n) and K = S(U(n) × U(1)), we see that

SU(n) × R −→ K (g, t) 7−→ eπitg, e−nπit (1) is the universal cover in coordinates on K. Note that 2 (e−πitI , t): t ∈ n nZ is the preimage of (In, 1) ∈ K in U(]n).

9 3.2 Coordinates on SU(n, 1) Consider G = SU(n, 1) as the group of special unitary automorphisms of n+1 C with respect to the hermitian form with matrix 0 0 1 h = 0 In−1 0 . 1 0 0

Fix the h-negative vector −1  0     .  z0 =  .     0  1 of h-norm −2 and let K be the stabilizer of in G of the line spanned by z0, i.e., the elements of G with z0 as an eigenvector. There is an isomorphism K =∼ S(U(n)×U(1)) determining coordinates (g, ξ) on K with g ∈ U(n) and −1 ξ = det(g) . In what follows, we identify SU(n) < K with g0 7→ (g0, 1) and −n U(1) < K with ξ 7→ (ξIn, ξ ). Then (g, ξ)z0 = ξz0 for all (g, ξ) ∈ K, hence SU(n) is the stabilizer of z0 in K and Kz0 = U(1)z0. The h-isotropic vector 1 0 v =   0 . . 0 has stabilizer containing the subgroup B0 = A+N < G of upper-triangular matrices with positive real eigenvalues. Speciﬁcally,

λ −λtz −λ|z|2/2 + it   n−1  B0 = 0 In−1 z  : λ ∈ R+, z ∈ C , t ∈ R  0 0 λ−1  where A+ is associated with the diagonal subgroup and N the unipotent radical of B0, which is isomorphic to the (n − 1)-dimensional Heisenberg group. We now prove that G = B0K, which allows us to ﬁx an identiﬁcation n of B0 = G/K with complex hyperbolic space B . In fact, we show that G = B0K gives an Iwasawa decomposition of G.

10 Lemma 3.1. With notation as above, G = B0K gives an Iwasawa decomposition of G with respect to the Cartan involution of G with matrix

0 0 1 Θ = 0 In−1 0 . 1 0 0

Proof. First, note that Θ = h, hence

tΘhΘ = h3 = h, and det(Θ) = −1, so Θ ∈ U(2, 1). If g denotes the Lie algebra of G, we claim that ad(Θ) ∈ End(g) is a Cartan involution with the Lie algebra a of 2 A+ as a Cartan subalgebra. Since Θ = In+1, Θ normalizes A+, and a is a maximal Ad-semisimple abelian subalgebra of g, to show that Θ is a Cartan involution ﬁxing the Cartan subalgebra a, it suﬃces to show that the +1 eigenspace for ad(Θ) is precisely the Lie algebra k of K. First, note that Θ is an element of the natural U(2) × U(1) < U(2, 1) containing K. Indeed, Θz0 = −z0. Moreover, Θ acts trivially on the h- orthogonal complement to z0, which is spanned by e1 + en+1 and e2, . . . , en, th where ei is the i standard basis vector. With respect to coordinates on U(2) × U(1) analogous to our coordinates on K, this means that

Θ = (In, −1).

It is clear from these coordinates that conjugation by Θ acts trivially on K, hence k is contained in the +1 eigenspace of ad(Θ). Conversely, if Θ commutes with g ∈ G, then g preserves the −1 eigenspace of Θ acting on n+1 C , which is the line spanned by z0. Thus z0 is an eigenvector of g, and so g ∈ K. This proves that ad(Θ) is a Cartan involution of g with +1 eigenspace k. Taking the root space decomposition of g with respect to a, we can choose positive roots so that the associated connected unipotent subgroup of G is N. Following [21, §VI.3], we see that B0K = A+NK is an Iwasawa decomposition of G. This completes the proof.

3.3 Coordinates on SU(^n, 1)

We now want to study the universal cover π : Ge → G, which ﬁts into a central extension: π 1 −→ Z −→ Ge −→ G −→ 1

11 where we identify Z with π1(G). First, we make the following observation, which follows immediately from the identiﬁcation of the center of G with Z/(n + 1). ∼ Lemma 3.2. If Z(Ge) denotes the center of Ge, then Z(Ge) = Z, π1(G) is naturally identiﬁed with (n + 1)Z(Ge), and Z(Ge) is naturally π1(PU(n, 1)).

We now lift our Iwasawa decomposition of G to one for Ge.

Lemma 3.3. With notation as above, let Ke be the universal cover of K. Then we have embeddings B , K → G so that B → G lifts B < G, π| is 0 e e 0 e 0 Ke the universal cover Ke → K, and B0Ke is an Iwasawa decomposition of Ge. Proof. This follows from the fact that π induces an isomorphism between the Lie algebras of Ge and G. Continuing with the notation from the proof of Lemma 3.1, Θ determines an Iwasawa decomposition of Ge with

A N = exp (an), e+ e Ge where n ⊂ g is the subalgebra associated with N. Since B0 is contractible and exp : an → B is a diﬀeomorphism, exp (an) =∼ B is a lift of B < G. G 0 Ge 0 0 Set K = exp (k), so we have an Iwasawa decomposition B K of G. It e Ge 0 e e remains to show that Ke is the universal cover of K with universal covering map π. Since B0 is contractible, we have π-equivariant diﬀeomorphisms B0\Ge → Ke and B0\G → K inducing homotopy equivalences of Ge with Ke and G with K. Since K is then connected and simply connected, π| must e Ke be the universal cover. This proves the lemma.

Applying the identiﬁcation of Ke with SU(n) × R, we have the following.

Corollary 3.4. There is an Iwasawa decomposition Ge = B0(SU(n) × R) compatible with the Iwasawa decomposition G = B0K and the universal cover SU(n) × R → K.

3.4 Useful homogeneous spaces

We now study a pair of homogeneous spaces for G and Ge that will end up being very helpful in computing lifts of relations in G up to Ge. Speciﬁcally, the homogeneous space G/ SU(n) is useful in calculating the element of ∼ 1 π1(G) = Z associated with a loop σ : S → G using only linear algebra.

12 n Lemma 3.5. Identifying G/K with B induces a diffeomorphism between n G/ SU(n) and B × U(1). The map g SU(n) 7→ gz0 moreover defines a left G-invariant diffeomorphism between G/ SU(n) and the space of all vectors n+1 in C of h-norm −2. n Proof. Identifying G/K with B , the Iwasawa decomposition G = B0K n identifies B0 with B . Then k = (h, ξ) 7→ ξ identifies K/ SU(n) with U(1), hence mapping b(h, ξ) ∈ B0K to (b, ξ) defines a diffeomorphism between n G/ SU(n) and B × U(1). This proves the first claim. The second claim is the orbit-stabilizer theorem, since SU(n) < K is the stabilizer of z0 and G acts transitively on vectors of h-norm −2.

Corollary 3.4 then gives a diagram of universal covers

' n G/e SU(n) B × R

' n G/ SU(n) B × U(1) and the long exact sequence of homotopy groups also implies that the ﬁbra- tion n G/ SU(n) B × U(1)

U(1) induces an isomorphism on the level of fundamental group. We can then compute elements of π1(G) as follows. Lemma 3.6. Given a loop σ : [0, 1] → G based at the identity of G, we obtain a loop σb based at 1 ∈ C by

σ [0, 1] G Gz0

πn+1 σb C

n+1 st where πn+1 is projection of C ⊃ Gz0 onto the (n + 1) coordinate. Then σb([0, 1]) ⊂ C r {0}. Identifying π1(G) with Z, the element [σ] ∈ π1(G) associated with σ is equal to the winding number of σb around 0 ∈ C.

13 Proof. For s ∈ [0, 1], let

σ(s) = bs(gs, ξs) ∈ B0K be the Iwasawa decomposition of σ(s). We then see that

 ∗   .   .  σ(s)z0 =   ,  ∗  −1 λs ξs

±1 where λs are the diagonal entries of bs with respect to the coordinates on B0 deﬁned in §3.2, hence

−1 πn+1(σ(s)z0) = λs ξs ∈ C r {0}.

Since σ is based at the identity of G, λ0 = ξ0 = 1, so σe is based at 1 ∈ C. This proves that σb([0, 1]) is a loop in C r {0} based at 1. n Recall that the identiﬁcation of Gz0 with B × U(1) in Lemma 3.5 iden- tiﬁes the U(1) factor with U(1)z0 for U(1) < K. This, with the previous calculation of πn+1 on Gz0, implies that the map

G Gz0

πn+1

C r {0} ∼ induces an isomorphism π1(G) = π1(C r {0}). The element of π1(C r {0}) associated with a loop is computed by the winding number around 0, so this completes the proof of the lemma.

4 Lifting presentations

In this section, ﬁx a ﬁnitely presented group Γ < G = SU(n, 1). We assume ∼ for simplicity that Γ contains the center Z(G) = Z/(n + 1) of G, which is always the case when Γ is the pullback to SU(n, 1) of a lattice in PU(n, 1), and let zb be a generator for Z(G). Without loss of generality, we can assume zb is a generator for Γ, and so Γ is given by generators and relations

Γ = hg1, . . . , gd, zb | R1,..., Rei.

14 We now describe a procedure for presenting the central extension

1 −→ Z −→ Γe −→ Γ −→ 1 obtained by pulling Γ back to the universal cover Ge of G. Lifting generators

Since G is connected, we can write gi = expG(vi) for some vi in the Lie algebra g of G. This deﬁnes an embedded path:

γi : [0, 1] −→ G γi(s) = expG(svi) −1 Notice that gi = expG(−vi). Then γ : [0, 1] −→ G γ (s) = exp (sv ) ei e ei Ge i is an embedded path in G with g = γ (1) a lift of g and g−1 = exp (−v ). e ei ei i ei Ge i Let z be a generator for the center of Ge that projects to z ∈ G. Then n+1 z is a generator for π1(G). Our generators for Γe are ge1,..., ged and z. Lifting relations

Now, let Ri be a relation for Γ. There is no loss of generality in considering Ri as being of the form g1,f ··· gi,1 = I ∈ G if i1 n+1 for i,j ∈ {±1} and f = f(i). This determines a piecewise-deﬁned loop σi : [0, 1] → G based at the identity by

 1 γi1 (i1 fs) s ∈ [0, ]  f σ (s) = γ ( (fs − j + 1))gi,j−1 ··· gi,1 s ∈ [ j−1 , j ] i ij i,j ij−1 i1 f f γ ( (fs − f + 1))gi,f−1 ··· gi,1 s ∈ [ f−1 , 1]  if i,f if−1 i1 f ∼ Using Lemma 3.6, we compute the element ri = [σi] ∈ Z = π1(G) deﬁned by σi using the winding number of the loop πn+1(σi(s)z0) around 0 in C r {0}. Notice that σ1(s) ∈ SLn+1(C), so the path σi(s)z0 is explicitly computed n+1 using matrix multiplication in C . Uniqueness of path lifting implies that the lift of σi to Ge based at the (n+1)r identity is given by z i for ri ∈ Z the winding number of πn+1 ◦ σi around 0 ∈ C. Thus our lifted relations Rei for Γe are: g 1,f ··· g i,1 z−(n+1)ri = 1 . eif ei1 e Ge

This presents Γe as the group with generators {gei} and relations {Rei}, pos- sibly with the additional relation that z is central.

15 Lemma 4.1. With notation as above,

Γe = hge1,..., ged, z | Re1,..., Ree, z centrali. Proof. Let L be the abstract group with the given presentation. The above shows that each relation for L holds in Γ,e hence we have a homomorphism L → Γ.e On the other hand, hzi is the kernel of the projection L → Γ, ∼ which factors through Γ.e Thus, the kernel of L → Γe is contained in hzi = Z. However, hzi maps isomorphically to the center of Γ,e so L =∼ Γ.e

5 Residual ﬁniteness of central extensions and the cup product

Let Γ be the fundamental group of a Riemann surface Σ of genus g ≥ 2. The fundamental group Γe of its unit tangent bundle T 1Σ fits into a central exact sequence 1 −→ Z −→ Γe −→ Γ −→ 1 2 ∼ 2 whose Euler class e ∈ H (Σ, Z) = H (Γ, Z) is the first Chern class of the canonical bundle. We can equivalently say that the Euler class of the bundle is a nonzero multiple of the Kählerform ω on Σ. Lastly, Γe is isomorphic to the preimage of Γ < PSL2(R) in its universal cover PSLg 2(R). It is well-known that Γe is residually finite, and the most common proof is to show that Γe admits a homomorphism onto an integral Heisenberg group such that a generator for the center of Γe maps to an element of infinite order. See [11, §IV.48] for details. It is not hard to prove by hand that integral Heisenberg groups are residually finite, and recall from §2 that in fact all finitely generated nilpotent groups are residually finite. Since Γ is linear, and hence residually finite, residual finiteness of Γe follows from Corollary 2.5. The purpose of this section is to present a broad generalization of this result, and its strategy of proof, based on a proposition that is perhaps best attributed to Sullivan (e.g., see [2, Ch. 3]). In what follows it will be convenient to adopt the convention that a k-step nilpotent group has length at most k. Our most general result is the following.

Theorem 5.1. Let X be a closed aspherical manifold with fundamental group Γ and U(1) −→ Y −→ X 2 be a principal U(1) bundle with Euler class ω ∈ H (X, Z) that has inﬁ- nite order. Suppose that cup product with ω deﬁnes an injective mapping

16 1 3 from H (X, Q) to H (X, Q). If Γe = π1(Y ) and z denotes a generator for π1(U(1)) < Γe, then the image of z in the maximal two-step nilpotent quotient of Γe has inﬁnite order if and only if ω is in the image of the map

^2 1 2 cQ : H (X, Q) → H (X, Q) given by evaluation of the cup product.

Remark 5.2. The aspherical assumption on X is only to conclude using the standard homotopy exact sequence that Γe is indeed a central extension of Γ by Z. Our methods also allow one to study the cases where any of our hypotheses on X and Y fail to hold, but we do not pursue these directions since this would take us too far aﬁeld from our applications to ball quotients.

Before embarking on the proof, we record some corollaries. The ﬁrst explains why Theorem 5.1 is exactly what we need in order to implement 1 the proof of residual ﬁniteness of π1(T Σ) for more general circle bundles over aspherical manifolds.

Corollary 5.3. With X, Y , Γ, and Γe as in the statement of Theorem 5.1, suppose that Γ is residually finite. Then residual finiteness of Γe can be detected in a two-step nilpotent quotient of Γe via Corollary 2.5 if and only if the Euler class of Y → X is in the image of the cup product map cQ. Proof. Since Γ is a finitely generated residually finite group, Corollary 2.5 says that it suffices find a nilpotent quotient of Γe where the generator z for π1(U(1)) maps to an element of infinite order. Taking the maximal two-step quotient of Γ,e the corollary then follows from Theorem 5.1. The motivation for our assumptions on the cup product with ω is that the hard Lefschetz theorem implies that these are properties of the Kähler form on a compact Kählermanifold of complex dimension at least two. Thus we have the following consequence of Theorem 5.1 and Corollary 2.5.

Corollary 5.4. Let X be an aspherical compact Kählermanifold with residually finite fundamental group for which the Kählerclass ω ∈ H2(X) is in V2 1 2 the image of the cup product map H (X, Q) → H (X, Q). If Y is a principal U(1) bundle over X with Euler class a nonzero multiple of ω, then π1(Y ) is residually finite. Proof. The case where X is a Riemann surface of genus g ≥ 2 is the classical case explained at the beginning of this section. When X is a torus, π1(Y ) is

17 a finitely generated nilpotent group, so again the result is known. If X has n 2n complex dimension n ≥ 2, then ω ∈ H (X, Q) is nontrivial, hence ω has 2 infinite order in H (X, Q). The hard Lefschetz theorem then implies that n−1 1 2n−1 φ 7→ φ ∧ ω defines an isomorphism between H (X, Q) and H (X, Q), 1 3 hence φ 7→ φ ∧ ω from H (X, Q) to H (X, Q) must be injective. Therefore, Corollary 5.3 applies to prove that π1(Y ) is residually finite. Note that Theorem 1.2 is the special case of Corollary 5.4 where X is a smooth compact ball quotient. Also recall that smooth compact ball quotients are smooth projective varieties, the real cohomology class of the Kählerform is always (up to normalization, of course) rational. Before proving Theorem 5.1, we define some notation and give some pre- liminary results. To start, we only assume that X is a closed manifold with fundamental group Γ. Let ∆ denote the derived subgroup of Γ. The earliest reference we could find to the following fact was in work of Sullivan [44], who commented that its proof follows from “a certain amount of soul searching classical algebraic topology”. Also see [2, Ch. 3]. We use a formulation given by Beauville [5]. Proposition 5.5 (Cor. 1(2) [5]). Let X be a connected space homotopic to a CW complex with fundamental group Γ and ∆ be the derived subgroup of Γ. Then there is a canonical exact sequence

2 ^ 1 cQ 2 1 −→ Hom(∆/[∆, Γ], Q) −→ H (X, Q) −→ H (X, Q) (2) with cQ the cup product map. We now relate the cup products on X and U(1) bundles over X via the Gysin sequence. Lemma 5.6. Let Y be a principal U(1) bundle with base X and Euler class 2 ω ∈ H (X, Z). Suppose that ω has infinite order and that cup product with 1 3 ω defines an injection H (X, Q) → H (X, Q). Then the class of the fiber in i ∼ i H1(Y, Q) is trivial, H (Y, Q) = H (X, Q) for i = 0, 1, and pullback under the 2 ∼ 2 projection π : Y → X induces an isomorphism H (Y, Q) = H (X, Q)/hωi. Moreover, the diagram

cX V2 1 Q 2 1 Hom(∆/[∆, Γ], Q) H (X, Q) H (X, Q)

π∗ π∗ π∗ cY V2 1 Q 2 ∼ 2 1 Hom(∆e /[∆e , Γ]e , Q) H (Y, Q) H (Y, Q) H (X, Q)/hωi (3)

18 commutes, where cX , cY are the cup product maps Γ, Γ are the fundamental Q Q e groups of X, Y , and ∆, ∆e are the derived subgroups of Γ, Γe.

Proof. Throughout this proof, all homology groups have Q coeﬃcients. Let i i+2 wi : H (X) → H (X) be the map induced by cup product with ω. Then the standard Gysin sequence for Y [39, Thm. 5.7.11] gives

∗ w 0 H1(X) π H1(Y ) π∗ H0(X) 0 H2(X) π∗

w ∗ H2(Y ) π∗ H1(X) 1 H3(X) π ···

∗ where π is pullback and π∗ is integration over the fiber. Then ω has infinite 2 2 order in H (X, Z) if and only if it is nonzero in H (X), which is moreover ∗ true if and only if w0 is injective. We conclude that π induces an isomorphism between Hi(X) and Hi(Y ) for i = 0, 1 when ω has infinite order in 2 1 0 H (X, Z). Therefore H (Y ) → H (X) is the zero map, and since this map is integration over the fiber, the class of the fiber in H1(Y ) must be trivial. 2 Further, if ω is infinite order in H (X, Z), then rank H2(X) ≤ rank H2(Y ) + 1 with equality if and only if w1 is injective. Since w1 is injective by hypothesis, we see that π∗ induces an isomorphism H2(X)/hωi → H2(Y ). Commutativity of the diagram in the statement of the lemma now follows from Proposition 5.5 and the fact that pullback is a ring homomorphism for the cup product. This completes the proof.

We are now prepared for the main technical result on two-step nilpotent quotients that we need for the proof of Theorem 5.1.

Lemma 5.7. Suppose that X, Y , Γ, and Γe satisfy the hypotheses of Theo- rem 5.1. Let ∆ (resp. ∆e ) be the derived subgroup of Γ (resp. Γe), and let z be a generator for ker(Γe → Γ). Then

dim(Hom(∆e /[∆e , Γ]e , Q)) = dim(Hom(∆/[∆, Γ], Q)) + (4) for ∈ {0, 1}. Moreover, = 1 if and only if the image of z in the maximal two-step nilpotent quotient of Γe has inﬁnite order. Proof. Let

N = Γ/[∆, Γ] Ne = Γe/[∆e , Γ]e

19 be the maximal two-step nilpotent quotients of Γ and Γ.e The derived subgroups of N and Ne are ∆/[∆, Γ] and ∆e /[∆e , Γ],e respectively. Let zb denote the image of z in Ne. Since z generates the fundamental group of the fiber, and the fiber is trivial in H1(Y, Q) by Lemma 5.6, the image of z in the abelianization of Γe must have finite order. Therefore, we conclude that there is some m ≥ 1 m so that zb ∈ ∆e /[∆e , Γ].e Indeed, Ne and Γe have the same abelianization and ∆e /[∆e , Γ]e is the commutator subgroup of Ne. We then have a diagram:

1 1 1

m 1 h zb i h zbi Z/m 1

1 ∆e /[∆e , Γ]e Ne Γe/∆e 1

1 ∆/[∆, Γ] N Γ/∆ 1

1 1 1

To justify this, one wants to know that N/e h zbi = N. Note that Γe → N/e h zbi factors through Γe/hzi = Γ, so N/e h zbi is a quotient of Γ. Moreover, ∆e projects to ∆ and [∆e , Γ]e has image [∆, Γ]. Thus the kernel of Γ → N/e h zbi is [Γ, ∆], i.e., the kernel of Γ → N, hence N = N/e h zbi. Then the vertical exact sequence in blue is an exact sequence of finitely generated abelian groups. Moreover, the free ranks of Γe/[∆e , Γ]e and Γ/[∆, Γ] m differ by ∈ {0, 1} with = 1 if and only if zb (equivalently, zb) has infinite order in Ne. These are equivalent to Equation (4) by taking Hom(−, Q), so this proves the lemma.

We are now prepared to prove Theorem 5.1.

Proof of Theorem 5.1. Let KX , IX , KY , IY denote the kernel and image of cX and cY , and k , i , k , i denote their dimensions over . Pullback Q Q X X Y Y Q 1 1 induces an isomorphism between H (X, Q) and H (Y, Q) by Lemma 5.6, ∗ and see from Equation (3) that π (IX ) = IY . Therefore we have equalities:

^2 1 kX + iX = dimQ H (X, Q)

20 ^2 1 = dimQ H (Y, Q)

= kY + iY In other words, iX − iY = kY − kX . (5)

Moreover, the proof of Lemma 5.6 implies that iX = iY if ω∈ / IX and iX = iY + 1 if ω ∈ IX . We conclude that the left side of Equation (5) equals 1 if and only if ω ∈ IX . Similarly, Equation (3) and Lemma 5.7 imply that the right side of Equation (5) is 1 if and only if the image of z in the maximal two-step nilpotent quotient of Γe has inﬁnite order. This proves the theorem.

We now give an example where Corollary 5.4 applies to prove residual finiteness of a lattice in the universal cover of PU(2, 1). Let ΓS < PU(2, 1) be the fundamental group of the Stover surface XS, a smooth compact ball quotient surface first studied in [41] and explored further by Dzambic and Roulleau [16]. In particular, Dzambic and Roulleau proved that the cup V2 1 2 product H (XS, Q) → H (XS, Q) is surjective, hence the canonical class is in the image of the cup product map. Corollary 5.4 then implies that the preimage of ΓS in the universal cover of PU(2, 1), which is the central extension with Euler class the canonical class, is residually finite. Since ΓS is commensurable with both the Cartwright–Steger surface and the Deligne– Mostow lattice with weights (11, 7, 2, 2, 2)/12 (see [41]), Lemma 2.1 allows us to conclude: Theorem 5.8. Let Γ < PU(2, 1) be either the Deligne–Mostow lattice with weights (11, 7, 2, 2, 2)/12, the fundamental group of the Cartwright–Steger surface, the fundamental group of the Stover surface, or any other lattice in PU(2, 1) commensurable with these. Then the preimage Γe of Γ in the universal cover of PU(2, 1) is residually finite. Remark 5.9. We cannot use Corollary 5.4 to prove Theorem 5.8 for the 1,0 Cartwright–Steger surface XCS directly. Since h (XCS) = 1, the image of the cup product from H1 to H2 is the one-dimensional subspace spanned 1,0 by δ = α ∧ α for some α ∈ H (XCS). Then δ ∧ δ = 0, hence δ cannot be a multiple of the canonical class and Corollary 5.4 does not apply. One can show with Magma that the derived subgroup of the maximal two-step nilpotent of the preimage of π1(XCS) in the universal cover is finite. Remark 5.10. Since, as described above, every finitely generated nilpotent group has a torsion-free subgroup of finite index, and torsion-free nilpotent

21 groups are linear, it isn’t hard to see that the same holds for an arbitrary ﬁnitely generated nilpotent group using the induced representation. More generally, linearity is a commensurability invariant of groups. Therefore, if we assume instead throughout this section that Γ = π1(X) is linear, then we conclude that Γe = π1(Y ) is also linear. Indeed, with notation as above, if Γ embeds in GLN1 (R) and Ne into GLN2 (R), the natural composition

Γe → Γ × N,e → GLN1+N2 (R) is a faithful linear representation of Γ.e Thus the results in this section hold with residual ﬁniteness replaced with linearity.

6 Complex projective surfaces with the same fundamental group as a ball quotient

The purpose of this section is to prove the following result. Theorem 6.1. Let M be a smooth complex projective surface such that π2(M) = {0} and π1(M) is isomorphic to a torsion-free cocompact lattice Γ 2 in PU(2, 1). Then M is biholomorphic to Γ\B . We start with some basic facts that we will need in the proof.

Lemma 6.2. If M is a smooth complex projective surface such that π1(M) is isomorphic to a torsion-free cocompact lattice Γ < PU(2, 1), then there 2 is a surjective holomorphic map f : M → Γ\B realizing the isomorphism ∼ π1(M) = Γ. 2 Proof. Set N = Γ\B . Then N is a K(Γ, 1) that realizes Γ as a Poincar´e duality group of dimension four and there is a nonconstant continuous map f : M → N realizing this isomorphism on fundamental groups. Using standard arguments from the theory of harmonic maps and Siu Rigidity [38, Thm. 1], we can assume that f is harmonic, hence pluriharmonic. Speciﬁ- cally, it follows from [6, Thm. 7.2(b)] that there are three possibilities: 1. f(M) is a closed geodesic on N;

2. there is a Riemann surface C so that f factors as the composition

φ M C f ψ N

22 where φ : M → C is a surjective holomorphic map and ψ : C → N is harmonic;

3. f is surjective and holomorphic (or conjugate holomorphic, which can be safely ignored by a change of complex structure).

However, the ﬁrst two cases are impossible. Indeed, in each case the isomor- ∼ phism f∗ :Γ −→ Γ would factor through a surjective homomorphism onto a group of cohomological dimension at most two, which is absurd since Γ has cohomological dimension four. This proves the lemma.

Proposition 6.3. Suppose that M is a smooth complex projective surface with π1(M) isomorphic to a torsion-free cocompact lattice Γ < PU(2, 1). Then the universal cover Mf of M is holomorphically convex. Proof. This follows directly from work of Katzarkov and Ramachandran [24, Thm. 1.2], since Γ is one-ended and the natural inclusion of Γ into PGL3(C) is Zariski dense. More generally, the Shafarevich conjecture is known for smooth projective varieties with linear fundamental group by more recent work of Eyssidieux, Katzarkov, Pantev, and Ramachandran [17].

Remark 6.4. We sketch an elementary proof of Proposition 6.3. Suppose that f : M → N is the holomorphic map from Lemma 6.2. Consider the diagram fe 2 Mf B

πM πN M N f where Mf is the universal cover of M and fe is a f∗-equivariant lift of f to the universal covers, where f∗ : π1(M) → Γ is the induced homomorphism 2 and π(M), Γ act by covering transformations. Since B is holomorphically convex, to prove that Mf is holomorphically convex it suﬃces to show that fe is proper. However, properness of fe is an easy consequence of the fact that f∗ is an isomorphism and compactness of M,N. In more detail, using these facts it is easy to show: −1 2 Key claim: For every y0 ∈ N and ye0 ∈ πN (y0) ⊂ B , the map

−1 −1 πM : fe (ye0) −→ f (y0) is a homeomorphism.

23 This shows that fe-preimages of points are compact. A slight variation of the argument gives that fe-preimages of suﬃciently small closed disks are compact, from which properness follows easily. Proof of Theorem 6.1. With the notation established earlier in this section, consider the Cartan–Remmert reduction p : Mf → Y of the holomorphically convex space Mf. A priori Y could be complex dimension one or two. Since fe is surjective and factors through p, the composition 2 Mf → Y → B (6) 2 implies that Y has a surjective map to B , hence Y is two-dimensional. This implies that Mf is nondegenerate in the sense of [3, p. 500]. Thus, by [3, p. 508-509] it follows that H3(M,f Z) = H4(M,f Z) = {0}. For dimension reasons, Hi(M,f Z) = {0} for i > 4. If π2(M) = π2(Mf) = {0}, then Mf is a simply connected space with π2(Mf) = {0} and Hi(M,f Z) = {0} for all i ≥ 3. Successively applying the Hurewicz theorem, all homotopy groups of Mf are trivial, thus Mf is contractible and so M is aspherical. It follows that the map f : M → N is a homotopy equivalence, and consequently follows easily that f is biholomorphic; see [38, Thm. 8].

Remark 6.5. The hypothesis of Theorem 6.1 can be weakened to only assume that π2(M) is ﬁnite, since Gurjar proved that π2(M) is torsion-free when Mf is holomorphically convex [20, Thm. 1].

7 Realizing central extensions

2 Fix d ≥ 2 and suppose that N = Γ\B is a smooth compact ball quotient. Our goal is to construct a smooth projective surface M with fundamental group that ﬁts into a central exact sequence:

1 −→ Z/d −→ π1(M) −→ Γ −→ 1 (7) The known constructions all appear to be roughly equivalent to an “extrac- tion of roots”, where M is a branched cover of N. We now explain one such variant, sketched in [2, Ex. 8.15]. We begin by describing some auxiliary objects and establishing notation.

7.1 The projective space of a direct sum

If V is a complex vector space and v ∈ V, v 6= 0, [v] ∈ P(V ) denotes the line determined by v. If V1,V2 are complex vector spaces, v1 ∈ V1, v2 ∈ V2, v1 and

24 v2 not both 0, we write simply [v1, v2] for the element [(v1, v2)] ∈ P(V1 ⊕V2). ∼ ∼ 3 The case of interest for us will be V1 = V2 = C . Consider P(V1) and P(V2) as subsets of P(V1 ⊕ V2) in the standard way

[v1] 7→ [v1, 0] [v2] 7→ [0, v2] and the general element of P(V1 ⊕ V2) is therefore of the form [v1, v2] with v1 ∈ V1, v2 ∈ V2, not both 0, thus presenting P(V1 ⊕ V2) as a “join” of P(V1) and P(V2). If we set

U = {[v1, v2] ∈ P(V1 ⊕ V2): v1 6= 0 and v2 6= 0}, then we have a decomposition

P(V1 ⊕ V2) = U t P(V1) t P(V2) that we can visualize as in Figure 1.

P(V1)

[v1]

[v1, v2] U

[v2] P(V2)

Figure 1: P(V1 ⊕ V2) = U t P(V1) t P(V2)

This determines a ﬁbration

∗ C U [v1, v2] (8)

P(V1) × P(V2) ([v1], [v2]) where the ﬁber over ([v1], [v2]), which we denote by <[v1], [v2]>, is

∗ <[v1], [v2]> = {[λv1, µv2]: λ, µ ∈ C } (9)

25 in other words, the projective line joining the points [v1, 0], [0, v2], with these ∗ ∗ two points removed. This is isomorphic to (C × C ) /diagonal which is in ∗ turn is isomorphic to C , but in more than one way, since 0 and ∞ can be interchanged. To describe this ﬁbration in terms of familiar line bundles, let

Lj P(Vj) × Vj

P(Vj) be the tautological line bundles, j = 1, 2. Now, let Lj be the pullback of −1 Lj to P(V1) × P(V2) under the projection to P(Vj). Letting L denote the inverse of a line bundle and L∗ the line bundle minus its zero section, we have identiﬁcations

−1∗ −1 ∗ U = L1 ⊗ L2 = L1 ⊗ L2 , (10) which one can see directly in coordinates from the calculation h i h µ i hλ i λv , µv = v , v = v , v 1 2 1 λ 2 µ 1 2 for [vj] ∈ P(Vj) and λ, µ 6= 0. Specifically, the first equality identifies the point [λv1, µv2] over ([v1], [v2]) with the graph of the linear map µ v 7→ v ∈ Hom( , ) =∼ −1 ⊗ , 1 λ 2 L1 L2 L1 L2 and similarly with the second identification.

7.2 Finite maps to PN This section refines the well-known connection between very ample line bundles and maps to projective spaces. Critical for our purposes is that one can produce a finite map, and we include the proof of this fact for completeness. Lemma 7.1. Suppose that X is an n-dimensional smooth projective variety 2 and η ∈ H (X, Z) is the first Chern class of a very ample line bundle on X. Then there exists a finite map

n α : X −→ P ∗ 2 n so that α ([H]) = η, where [H] ∈ H (P , Z) is the Poincar´edual to a hyperplane.

26 N N Proof. Let α0 : X → P , N > n, be an embedding of X in P determined by a basis for the space of global sections of the very ample line bundle. ∗ 2 N Then α0([H0]) = η, where [H0] ∈ H (P , Z) is the hyperplane class. See [19, §1.4] for a standard reference. N Now, choose p0 ∈ P so that p0 ∈/ α0(X) and the generic hyperplane N −1 H0 ⊂ P though p0 is transverse to α0(X). Then α0 (H0) is Poincar´edual ∗ to α0([H0]) = η. Let N N−1 π0 : P r {p0} −→ P be projection centered at p0 and α1 = π0 ◦ α0. Then, by construction, α1 is −1 a ﬁnite map. Since π0 determines a correspondence between hyperplanes N−1 N in P and hyperplanes in P through p0, we have that −1 −1 α1 (H1) = α0 (H0)

N−1 −1 for all hyperplanes H1 ⊂ P , where H0 = π0 (H1). Thus, for a general N−1 −1 ∗ H1 ⊂ P , we see that α1 (H1) is Poincarédual to η, i.e., α1([H1]) = η. N−j We now iterate this procedure to obtain αj : X → P . As long as N − j > n, we can find a point pj ∈/ αj(X) that allows us to construct αj+1. Note that the composition of finite maps is finite, so each αj is a finite map. n This ends with α = αN−n : X → P , which is the desired map.

7.3 The construction of M We now build the smooth projective surface M described at the beginning 2 of this section. Let N = Γ\B be a smooth compact ball quotient. Fix a very ample line bundle on N, e.g., an appropriate power of the canonical bundle, and let 2 α : N → P be the map provided by Lemma 7.1. If H denotes a generic hyperplane 2 ∗ on P , then α ([H]) is the ﬁrst Chern class of our chosen very ample line bundle. Finally, let ∗ C Z

N ∗ denote the C bundle over N obtained by removing the zero section from the total space of our chosen very ample line bundle, and set Γe = π1(Z). 2 ∼ 2 Note that the element of H (Γ, Z) = H (N, Z) associated with realizing Γe as a central extension of Γ by Z is the same as the ﬁrst Chern class of our line bundle.

27 2 2 th Fix d ≥ 2, and let β : P → P be the d power map

d d d [x0 : x1 : x2] 7→ [x0 : x1 : x2]. (11)

∗ 2 Note that β ([H]) = d[H], and that β is a (Z/d) branched cover. From §7.1, we have the ﬁbration

F 5 V U P (12)

2 α×β 2 2 N × P P × P

2 ∗ 2 2 where V is the pullback to N × P of the C bundle U over P × P . More precisely,

2 V = (x, y, u): x ∈ N, y ∈ P , u ∈ <α(x), β(y)> , (13) where <α(x), β(y)> is as deﬁned in Equation (9) and

F (x, y, u) = u (14) for (x, y, u) ∈ V . The following theorem is a very special case of [18, Thm. II.1.1]. Specif- ically, it follows from the quoted theorem of Goresky and MacPherson by checking that, in our case, their nb reduces to n. N Theorem 7.2. Let V be a smooth algebraic variety and f : V → P be an N algebraic map with ﬁnite ﬁbers. If Λ ⊂ P is a generic linear subspace, then −1 πi(V, f (Λ)) = 0 for i ≤ dim(Λ).

5 We will apply Theorem 7.2 to our F : V → U ⊂ P deﬁned in Equa- 2 tion (12) and Λ ⊂ U a generic P . First, observe that any 2-dimensional projective subspace 5 Λ ⊂ U ⊂ P is disjoint from P(V1) and P(V2). Consequently, it is the graph of a linear isomorphism A : V1 → V2, that is,

Λ = {[v1, Av1]: v1 ∈ V1} .

We now describe F −1(Λ) more precisely. −1 2 ∼ ∗ To ﬁnd F (Λ), note that for any (x, y) ∈ N × P , <α(x), β(y)> = C can meet Λ at at most one point. Indeed, otherwise <α(x), β(y)> ⊂ Λ, and

28 Λ would then contain the points in each of P(V1) and P(V2) contained in the closure of <α(x), β(y)>, contradicting the fact that Λ ⊂ U. Moreover, <α(x), β(y)> ∩ Λ 6= ∅ if and only if there exist αg(x) ∈ V1 and βg(y) ∈ V2 in the lines α(x) ∈ P(V1) and β(y) ∈ P(V2), respectively, such that βg(y) = Aαg(x). This is the same as saying that β(y) = Aα(x), where A : P(V1) 7→ P(V2) is the map of projective spaces induced by A. This is moreover equivalent to saying α(x) = A−1β(y). In other words, if we deﬁne

2 M = (x, y) ∈ N × P : β(y) = Aα(x) (15) 2 −1 = (x, y) ∈ N × P : α(x) = A β(y) we have seen that <α(x), β(y)> ∩ Λ 6= ∅ if and only if (x, y) ∈ M. Con- versely, if (x, y) ∈ M then there exits a unique point u(x, y) contained in <α(x), β(y)> ∩ Λ. Therefore,

F −1(Λ) = {(x, y, u(x, y)) ∈ V :(x, y) ∈ M} = Mc and F −1(Λ) ⊂ V . From now on we denote F −1(Λ) by Mc, which is the image ∗ 2 of a section over M of the C -bundle V → N × P of Equation (12). In particular, Mc is biholomorphic to M. In terms of the diagram in Equation (12), M and Mc ﬁt as

−1 F 5 Mc = F (Λ) V U P (16)

2 α×β 2 2 M N × P P × P where the left-most vertical arrow is a biholomorphism. Note that Λ, M, and Mc depend on the linear isomorphism A : V1 → V2. Lemma 7.3. With notation as above, given a generic linear isomorphism −1 A : V1 → V2, the complex surfaces Mc = F (Λ) ⊂ V and its projection 2 M ⊂ N × P deﬁned in Equation (15) are biholomorphic smooth projective varieties. Further, M can be described as a ﬁber product by either of the diagrams M P(V2) M P(V2)

A−1β β α Aα N P(V1) N P(V2) 2 in which the vertical maps are (Z/d) -branched covers.

29 −1 2 Proof. We already proved that Mc = F (Λ) ⊂ V projects to M ⊂ N ×P as in the diagram from Equation (16) and that Mc is the image of a section over 2 M of the bundle V → N × P , hence the projection is biholomorphic. By its deﬁnition from Equation (15), M is a closed subvariety of the projective 2 variety N ×P , hence it is projective and the squares are ﬁber-squares. Since 2 β is a branched (Z/d) -cover by Equation (11), so are the other vertical maps. So far, the above is true for any A. It remains to prove that, for generic 2 A, M is a smooth subvariety of N × P . To see this, the map 2 2 2 GL3(C) × N × P −→ N × P × P f (A, x, y) 7−→ (x, Aα(x), y) is a submersion. Therefore, Sard’s theorem implies that the map fA(x, y) = (x, Aα(x), y) is transverse to the submanifold

2 2 2 Y = (x, β(y), y): x ∈ N, y ∈ P ⊂ N × P × P

2 for almost every A ∈ GL3(C). Then fA(N × P ) ∩ Y is precisely the above ﬁber product associated with the given choice of A, hence we see that the ﬁber product is a manifold for almost every choice of A. Said otherwise, M is a manifold for almost every Λ. This completes the proof.

We now show that π1(M) fits into the desired central exact sequence involving Γ. We will also compute π2(M) under the hypothesis that π1(M) is residually finite, which we will use to show that M is not a modification of a ball quotient. Theorem 7.4. With notation as above, the smooth projective surface M has the following properties. 1. The fundamental group of M fits into a central exact sequence:

1 −→ Z/d −→ π1(M) −→ Γ −→ 1 (17)

2. The exact sequence in Equation (17) is the reduction modulo d of the central exact sequence

1 −→ Z −→ Γe −→ Γ −→ 1 (18)

for the fundamental group Γe of Z.

3. If π1(M) is residually ﬁnite, then π2(M) 6= {0}.

30 Proof of parts 1 and 2 of Theorem 7.4. Since α and β are finite maps and F : V → U is a bundle map (that is an isomorphism on the fibers of the 5 fibrations in Equation (12)), it follows that the map F : V → U ⊂ P has finite fibers. Thus Theorem 7.2 applies and gives us πi(V, Mc) = 0 for i ≤ 2. The homotopy exact sequence

∼ π2(Mc) π2(V ) π2(V, Mc) = {0}

∼ π1(Mc) π1(V ) π1(V, Mc) = {0}

∼ π0(Mc) π0(V ) = {0}

∼ then implies that Mc is connected, π1(Mc) = π1(V ), and π2(Mc) maps onto π2(V ). In the last three statements, we can replace Mc by M. We therefore want ∗ to compute the homotopy groups of V . Considering V as a C ﬁbration over 2 N × P , we have a the exact sequence:

∗ ∼ 2 ∼ 2 ∼ π2(C ) = {0} π2(V ) π2(N × P ) = π2(P ) = Z

∗ ∼ 2 ∼ π1(C ) = Z π1(V ) π1(N × P ) = π1(N) {0}

2 ∗ Once we understand the map π2(P ) → π1(C ), computing πi(V ) for i ≤ 2 will be direct. 2 For a ﬁxed y ∈ N and z ∈ P , V restricts to ﬁbrations isomorphic to:

∗ ∗ −1∗ ∗ ∗ −1∗ C α L1 C β L2

2 N × {z} {y} × P

∗ To simplify notation, let L1 and L2 denote the total spaces of these C 2 ∗ bundles over N × {z} and {y} × P , respectively. Since β L2 is the line bundle OP2 (−d) by construction, L2 is OP2 (d) minus its zero section. Since

31 ∗ π2(C ) = {0}, we have homomorphisms of exact sequences

{0} Z Γ ∗ {0} π2(L1) π2(N) π1(C ) π1(L1) π1(N) {0}

∼= ∼=

2 ∗ 2 {0} π2(V ) π2(N × P ) π1(C ) π1(V ) π1(N × P ) {0}

∼= ∼= {0} π (L ) π ( 2) π ( ∗) π (L ) π ( 2) {0} 2 2 2 P ×d 1 C 1 2 1 P {0} Z Z Z/d {0} (19) where the bottom row is the standard homotopy exact sequence for the complement of the zero section in OP2 (d). Following the blue portion of Equation (19), we see that π1(V ) ﬁts into an exact sequence:

1 −→ Z/d −→ π1(V ) −→ Γ −→ 1 (20) From the orange portion, we see that Equation (20) is the reduction modulo d of the exact sequence

1 −→ Z −→ π1(L1) −→ Γ −→ 1 (21) for the fundamental group of the total space of the line bundle over N with ∼ ﬁrst Chern class η minus its zero section. Since π1(V ) = π1(M), this proves parts 1 and 2 of the theorem.

Remark 7.5. The above argument shows that π2(M) surjects π2(V ), but that π2(V ) is trivial. Thus we cannot conclude directly from V that π2(M) is nontrivial. With parts 1 and 2 of Theorem 7.4, we can prove Theorem 1.3.

Proof of Theorem 1.3. Let Γ < PU(2, 1) be a torsion-free lattice and Γe be its preimage in the universal cover. Fixing d ≥ 2 and applying parts 1 and 2 of Theorem 7.4, we obtain a smooth projective surface Sd so that there is an exact sequence

1 −→ Z/d −→ π1(Sd) −→ Γ −→ 1 where the center of π1(Sd) is obtained by reducing the center of Γe modulo d. This central extension is precisely the preimage of Γ in the d-fold connected cover of PU(2, 1). Since π1(Sd) contains elements of ﬁnite order, and

32 fundamental groups of smooth ball quotients are torsion-free, we see that Sd cannot be birational to a ball quotient. This proves the theorem. We now complete the proof of Theorem 7.4.

Proof of part 3 of Theorem 7.4. If π1(M) is residually finite, Lemma 2.2 implies that there is a finite index subgroup ∆ of π1(M) that is isomorphic to a subgroup of Γ. This implies that there are finite étalecovers M 0 → M and N 0 → N such that the finite map f : M → N defined by the fibre squares the statement of Lemma 7.3 lifts to a map fb : M 0 → N 0 that induces an isomorphism on fundamental groups. 0 ∼ Note that π2(M ) = π2(M). If π2(M) was zero (even finite), then The- orem 6.1 implies that M 0 is a ball quotient, so in particular it is aspherical. The map fb : M 0 → N inducing an isomorphism of fundamental groups would then be a homotopy equivalence, in particular have degree ±1. However, if 00 M is the cover of M associated with the subgroup Z/d × ∆ containing ∆, we obtain a diagram of maps

deg d2 M N

M 00 N 0

d-to-1 fb M 0

This shows that fb : M 0 → N has degree d3 > 1, contradicting the previous calculation of this degree. This contradiction implies that π2(M) must be nontrivial (in fact, inﬁnite), completing the proof of the theorem.

To give a concrete example of how one can use the methods of this section, we now prove Theorem 1.4.

Proof of Theorem 1.4. Let Γ be the fundamental group of a fake projective plane X whose canonical bundle KX is divisible by three. The case d = 3 of Theorem 7.4 applied to the preimage Γe of Γ in the universal cover of PU(2, 1) provides a smooth complex surface S whose fundamental group is the central Z/3 extension of Γ isomorphic to the preimage of Γ in SU(2, 1). ∼ Our assumption on KX implies that this lift splits, so π1(S) = Z/3×Γ (e.g., see [27, §8.9]). It follows that there is a degree 3 ´etalecover S0 → S with fundamental group Γ. By construction, S0 admits a degree 9 map onto X, hence there

33 is a degree 27 map from S0 to X. Arguing as in the proof of part 3 of Theorem 7.4, this implies that S0 cannot be a modiﬁcation of X. Thus S0 is the desired surface.

Remark 7.6. We briefly explain how one can determine the Chern numbers of the surfaces in Theorem 1.4. Let X be a fake projective plane such that KX is divisible by 3. We need to replace KX with a very ample line bundle with the same first Chern class as KX modulo 3. Using Reider’s theorem as in [8, Thm. 2.1], one easily seems that nKX is very ample for all n ≥ 3, and it has the same Chern class modulo 3 as KX , since KX is divisible by 3. In fact, Catanese and Keum prove that one can take 2KX for many fake projective planes [8, Thm. 1.1]. Proceeding with our construction, we take Sn to be the surface with fundamental group π1(X) × Z/3 built above. The construction of Sn as a 2 fiber product implies that it is the (Z/3) branched cover of X over three general members D1,D2,D3 of the linear system |nKX |. Note that “general” here is only needed to ensure that the triple intersection is empty. We compute the genus of Dj by adjunction:

2 2(g − 1) = KX Dj + Dj 2 2 2 = nKX + n KX = 9n(n + 1)

2 Each Dj meets another of the curves Dk in DjDk = 9n points, hence each 2 2 Dj contains 18n singular points and there are 27n singular points in total. Branching of Sn → X has order 3 over each nonsingular point of Dj and order 9 over the singular points. For example, following [4, p. 16-17], we can then compute the Chern numbers of Sn by:

1 2 c2(S ) = 9 K + 3 1 − nK 1 n X 3 X 2 = 27 ((2n + 1)KX ) = 243 (2n + 1)2 1 1 c (S ) = 9 3 − 3 1 − (−9n(n + 1) − 18n2) − 27n2 1 − 2 n 3 9 = 81 (10n2 + 6n + 1)

0 0 ∼ Now, let Sn be the ´etale Z/3 cover of Sn with π1(Sn) = π1(X) < π1(Sn). 0 Note that the Chern numbers of Sn are three times those of Sn. This

34 0 0 0 determines a sequence of surfaces Sn, n ≥ 1, where S1 = X, maybe S2 is not deﬁned if 2KX is not very ample, and for n 6= 1 we have:

3(2n + 1)2 n→∞ 6 c2(S0 )/c (S0 ) = −−−→ 1 n 2 n 10n2 + 6n + 1 5

Note that Troncoso and Urz´ua[47] produce examples of surfaces Yα with ∼ 2 π1(Yα) = π1(X) and c1(Yα)/c2(Yα) dense in [1, 3], whereas we only obtain the limit point 1.2.

8 Examples

This section provides the examples necessary to complete the proof of The- orem 1.1 using presentations of preimages in the universal cover. In particular, we use the method developed in §4. Our examples are all commensurable with the lattices constructed by Deligne and Mostow [13, 32]. Throughout th this section, ζn will denote a primitive n root of unity. We study Deligne–Mostow lattices Γ < PU(2, 1) for which the underlying 2 space for the orbifold Γ\B is the weighted projective space P(1, 2, 3), i.e., 2 the quotient of P by the natural action of S3 by coordinate permutations. A basic reference for the underlying spaces for Deligne–Mostow orbifolds is [26, Thm. 4.1]; in their notation, we are interested in the cases where Q(µ) 2 is P and µ has symmetry group S3. We begin with some basics on a standard presentation for these lattices based on the topology of P(1, 2, 3). If [x1 : x2 : x3] denotes homogeneous 2 coordinates on P , let A be the image on P(1, 2, 3) of the lines xi = xj and B denote the images of the coordinate lines xi = 0. Then A and B are tangent at one point, intersect transversally at another, B has a cusp singularity, there is a singularity of P(1, 2, 3) of type A1 on A, and a singularity of type A2 that lies on neither A or B. See Figure 2 for a visualization. Consider the loops b, u, and v around A and B pictured in Figure 2. One can show that

3 π1(P(1, 2, 3) r {A, B}) = b, u, v | br2(b, v), br3(b, u), br4(u, v), (buv) , where brm(a, b) denotes the braid relation of length m: ( ab ··· ab = ba ··· ba m even brm(a, b) = (22) ab ··· ba = ba ··· ab m odd

35 A B

v u

b × A1 A2 ×

Figure 2: A visualization of P(1, 2, 3)

It follows from standard facts about orbifolds and the procedure described in [13, 32] for computing orbifold weights that we obtain the presentation

r1 r1 r2 3 Γ = b, u, v | b , u , v , br2(b, v), br3(b, u), br4(u, v), (buv) where r1 and r2 are computed directly from the set of weights µ. At the two crossings between A and B, as well as at the singularity of B, the local orbifold group is then a complex reﬂection group

p q p[m]q = ha, b | a , b , brm(ab)i generated by complex reflections of order p and q. When p[m]q is finite (i.e., it is spherical), it admits a standard representation to U(2) where det(a) = ζp and det(b) = ζq, each with eigenvalues 1 and ζr for r = p, q. See [10, §9.8]. When p[m]q is infinite it is Euclidean and the associated point of 2 P(1, 2, 3) is a cusp point of the complex hyperbolic orbifold, that is, Γ\B is the complement of the cusp points in P(1, 2, 3). Example 1: (5,4,1,1,1)/6 (nonuniform, arithmetic) This example is noncompact, arithmetic, and commensurable with the Picard modular group over the field Q(ζ3). The curve A in Figure 2 has orbifold weight 6 and the curve B has weight 3. The point of tangency between A and B is a cusp with Euclidean cusp group 3[4]6. See the appendix to [43] for more on the orbifold structure and details concerning the computations that follow. Our presentation is:

3 3 6 3 Γ = b, u, v | b , u , v , br2(b, v), br3(b, u), br4(u, v), (buv)

36 With respect to the standard hermitian form h in §3.2, we can choose generators for Γ < PU(2, 1) with the following representatives in U(2, 1): √    5  1 0 −1/ −3 √ζ6 0 0 5 b0 7→ √0 ζ6 0  u0 7→ √−3 √ζ6 0  5 −3 0 0 −3 −3 ζ6   ζ6 0 0 v0 7→  0 ζ3 0  0 0 ζ6

Scaling these by the cube root of their determinant, we obtain elements bb, u,b vb ∈ SU(2, 1) that generate the preimage Γb of Γ in SU(2, 1). Our generators in SU(2, 1) satisfy the relations

3 3 6 2 b = ub = vb = zb 3 (bubvb) = Id where zb = ζ3 Id is the standard generator for the center of SU(2, 1). More- over, one checks that br2(bb, vb), br3(bb, ub), br4(u,b vb) all hold. The above relations give a presentation for Γ.b Now, using the methods developed in §4, we can ﬁx lifts b, u, and v of our generators of Γ to the universal cover Ge of SU(2, 1) and use computational software to compute the associated lift of each relation. For example, see Figures 3-5 for a graphical representation produced in Mathematica of the projection to C r {0} as in §4 of the loop in SU(2, 1) given by the relation b9 = Id with b3 a generator for the center of SU(2, 1). Working through the lift of each relation, one obtains:

Theorem 8.1. Let Γ < PU(2, 1) be the Deligne–Mostow lattice with weights (5, 4, 1, 1, 1)/6. The preimage Γe of Γ in the universal cover Ge has presentation

3 3 6 3 3 b, u, v, z : b z, u z, v z, br2(b, v), br3(b, u), br4(u, v), (buv) z , where z is a lift to Ge of a generator for the center of SU(2, 1) and hence z3 denotes a generator for π1(SU(2, 1)).

We now explain how one proves that Γe is residually finite. We exploit the subgroup of index 72 associated with the ball quotient manifold constructed by Hirzebruch [23]. Briefly, this is the fundamental group of one of the five ball quotient manifolds of Euler number 1 that admit a smooth toroidal

37 b 0.0 0.7 0.8 0.9 1.0

-0.5

-1.0

-1.5

Figure 3: Lifting b starts turning clockwise from 1 ∈ C r {0}.

b^3 with ray toE^(4*Pi*I/3)

-0.4 -0.2 0.2 0.4 0.6 0.8 1.0

-0.5

-1.0

-1.5

-2.0

Figure 4: The path for the lift of b3 ends at e4πi/3, so b3 lifts to z−1. compactiﬁcation [40, 15]. See [42, 14] for more on the geometry of this manifold and further applications to our understanding of the structure of ball quotients and their fundamental groups. Let Λ < Γ be the fundamental group of Hirzebruch’s ball quotient. See [43, Prop. A.8] for details concerning how one can ﬁnd Λ, say in Magma, as a normal subgroup of Γ with index 72. Let Λe be the preimage of Λ in Γ,e N be the maximal two-step nilpotent quotient of Λ, and Ne the maximal two-step nilpotent quotient of Λ.e Using Magma, we see that there are exact sequences

3 4 1 −→ Z −→N −→ Z −→ 1 4 4 1 −→ Z −→Ne −→ Z −→ 1

38 b^9- full circle 1.5

1.0

0.5

-2.0 -1.5 -1.0 -0.5 0.5 1.0 1.5

-0.5

-1.0

-1.5

-2.0

Figure 5: A full clockwise rotation at b9 = Id. and that z ∈ Γe maps to an inﬁnite order element of the derived subgroup of Ne. Corollary 2.5 then implies that Λe is residually ﬁnite, hence Lemma 2.1 gives the following.

Theorem 8.2. Let Γ < PU(2, 1) be any lattice commensurable with the Picard modular group over the field Q(ζ3) and Γe be its preimage in the universal cover of PU(2, 1). Then Γe is residually finite and linear. Lattices that fall under the umbrella of Theorem 8.2 include the Deligne– Mostow lattices with weights (2, 1, 1, 1, 1)/3, (5, 4, 1, 1, 1)/6, and seven oth- ers. It also covers the fundamental group of Hirzebruch’s ball quotient and in fact every minimal volume complex hyperbolic 2-manifold that admits a smooth toroidal compactification [15]. Example 2: (11,7,2,2,2)/12 (uniform, arithmetic) This example is already covered by Theorem 5.8, and the details from this perspective are very similar to the previous example, so we are somewhat terse. We include it here because these methods give some more precise information about the relationships between the two-step nilpotent quotients of the fundamental group of the Stover surface and the associated central extension. For this example, the curve A has orbifold weight 4 and B again has weight 3. For the hermitian form with matrix  √  1 + 3 −1√ 0  −1 −1 + 3 0  0 0 −1

39 we can choose the following representatives in U(2, 1) for the generators:

 1 0 0  √ √ √ √ b = ( 3 + 3) − (1 + 3 )i 1 (−1 + i)(1 + 3) − i (2 − i + 3) 0  2 √ 2 2 2 √  1 1 2 (1 + i)(1 + 3) −1 − i 2 (2 − i + 3) √ √  1 (1 + i)(1 + −3) 1 − 3+i 0 2 √ √ 2 u = i 1 0  2 (2 + i + 3) 2 ( 3 − i) 0 0 0 1   i √ 0 0 1 v0 =  2 (−1 + i)(1 + 3) 1 0 0 0 1

To obtain elements of U(2, 1) with respect to the standard hermitian form used in §3.2, we conjugate by the matrix q √ p √  1 (1 + 3) − 1 −1 + 3 − √1 2 2 2  1   0 √ √ 0   1+ 3  q √ p √  1 (1 + 3) − 1 −1 + 3 √1 2 2 2

1/3 and then scale by det to obtain elements bb, u,b vb ∈ SU(2, 1) with respect to our standard hermitian form. Similarly to the previous example, we obtain:

Theorem 8.3. Let Γ < PU(2, 1) be the Deligne–Mostow lattice with weights (11, 7, 2, 2, 2)/12. The preimage Γe of Γ in the universal cover Ge has presentation

3 3 4 3 3 b, u, v, z : b z, u z, v z, br2(b, v), br3(b, u), br4(u, v), (buv) z , where z is a lift to Ge of a generator for the center of SU(2, 1) and hence z3 denotes a generator for π1(SU(2, 1)). Remark 8.4. One can reinterpret Theorems 8.1 and 8.3 as follows. With q equal to 6 or 4, respectively, Γe is generated by the natural preimages in Ge of the ﬁnite complex reﬂection subgroups

3[2]q = hb, vi 3[3]3 = hb, ui 3[4]q = hu, vi

3 of PU(2, 1), subject to the relation that (buv) is a generator for π1(SU(2, 1)). Here b, u, v denote generators in PU(2, 1) and b, u, v are lifts to the universal cover.

40 Using Theorem 8.3, we now give a second proof of Theorem 5.8.

Second proof of Theorem 5.8. Consider the fundamental group of the Stover surface, which is the congruence subgroup Λ ≤ Γ of prime level dividing 3 (see [41, 16]). This is a normal subgroup of index 18144 for which the preimage of Λ in SU(2, 1) splits, i.e., is isomorphic to Λ × Z/3. Considering Λ as a torsion-free lattice in SU(2, 1), we have a preimage Λe < Ge of Λ of the form

1 −→ hz3i −→ Λe −→ Λ −→ 1 with Λe a subgroup of index 54432 in Γe and, as above, z3 is a generator for π1(SU(2, 1)). Using Magma, one can compute the maximal two-step nilpotent quotients N and Ne of Λ and Λ,e respectively. These ﬁt into exact sequences:

28 14 1 −→ Z/4 × Z −→N −→ Z −→ 1 29 14 1 −→ Z −→Ne −→ Z −→ 1

3 ∼ th Then hz i = Z maps injectively to Ne, and is generated by the 4 power of 29 a certain primitive element of Z . As in previous examples, Corollary 2.5 implies that Λe is residually ﬁnite and Lemma 2.1 completes the proof of the theorem.

References

[1] Trento examples. In Classification of irregular varieties (Trento, 1990), volume 1515 of Lecture Notes in Math., pages 134–139. Springer, 1992. URL: https://doi-org/10.1007/BFb0098342. [2] J. Amorós,M. Burger, K. Corlette, D. Kotschick, and D. Toledo. Fun- damental groups of compact Kählermanifolds, volume 44 of Mathemat- ical Surveys and Monographs. American Mathematical Society, 1996. URL: https://doi-org/10.1090/surv/044. [3] A. Andreotti and R. Narasimhan. A topological property of Runge pairs. Ann. of Math. (2), 76:499–509, 1962. URL: https://doi-org/ 10.2307/1970370. [4] G. Barthel, F. Hirzebruch, and T. Höfer. Geradenkonfigurationen und Algebraische Flächen. Aspects of Mathematics, D4. Friedr. Vieweg & Sohn, 1987. URL: https://doi.org/10.1007/978-3-322-92886-3.

41 [5] A. Beauville. On the second lower quotient of the fundamental group. In Algebraic and complex geometry, volume 71 of Springer Proc. Math. Stat., pages 41–45. Springer, 2014. URL: https://doi-org/10.1007/ 978-3-319-05404-9_2.

[6] J. A. Carlson and D. Toledo. Harmonic mappings of K¨ahlermanifolds to locally symmetric spaces. Inst. Hautes Etudes´ Sci. Publ. Math., (69):173–201, 1989. URL: http://www.numdam.org/item?id=PMIHES_ 1989__69__173_0.

[7] D. I. Cartwright and T. Steger. Enumeration of the 50 fake projective planes. C. R. Math. Acad. Sci. Paris, 348(1-2):11–13, 2010. URL: https://doi-org/10.1016/j.crma.2009.11.016.

[8] F. Catanese and J. Keum. The bicanonical map of fake projective planes with an automorphism. Int. Math. Res. Not. IMRN, (21):7747–7768, 2020. URL: https://doi.org/10.1093/imrn/rny214.

[9] L. Clozel. Produits dans la cohomologie holomorphe des vari´et´esde Shimura. II. Calculs et applications. J. Reine Angew. Math., 444:1–15, 1993. URL: https://doi-org/10.1515/crll.1993.444.1.

[10] H. S. M. Coxeter. Regular complex polytopes. Cambridge University Press, second edition, 1991.

[11] P. de la Harpe. Topics in geometric group theory. Chicago Lectures in Mathematics. University of Chicago Press, 2000.

[12] P. Deligne. Extensions centrales non résiduellement finies de groupes arithmétiques. C. R. Acad. Sci. Paris Sér.A-B, 287(4):A203–A208, 1978.

[13] P. Deligne and G. D. Mostow. Monodromy of hypergeometric func- tions and nonlattice integral monodromy. Inst. Hautes Etudes´ Sci. Publ. Math., (63):5–89, 1986. URL: http://www.numdam.org/item? id=PMIHES_1986__63__5_0.

[14] L. F. Di Cerbo and M. Stover. Multiple realizations of varieties as ball quotient compactiﬁcations. Michigan Math. J., 65(2):441–447, 2016. URL: https://doi-org/10.1307/mmj/1465329021.

[15] L. F. Di Cerbo and M. Stover. Classiﬁcation and arithmeticity 2 of toroidal compactiﬁcations with 3c2 = c1 = 3. Geom. Topol.,

42 22(4):2465–2510, 2018. URL: https://doi-org/10.2140/gt.2018. 22.2465.

[16] A. Dˇzambi´cand X. Roulleau. On the cohomology of the Stover surface. Exp. Math., 26(4):490–495, 2017. URL: https://doi-org/10.1080/ 10586458.2016.1205533.

[17] P. Eyssidieux, L. Katzarkov, T. Pantev, and M. Ramachandran. Linear Shafarevich conjecture. Ann. of Math. (2), 176(3):1545–1581, 2012. URL: https://doi-org/10.4007/annals.2012.176.3.4.

[18] M. Goresky and R. MacPherson. Stratiﬁed Morse theory, volume 14 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3). Springer, 1988. URL: https://doi-org/10.1007/978-3-642-71714-7.

[19] P. Griﬃths and J. Harris. Principles of algebraic geometry. Wiley Classics Library. John Wiley & Sons, 1994. Reprint of the 1978 original. URL: https://doi-org/10.1002/9781118032527.

[20] R. V. Gurjar. Two remarks on the topology of projective surfaces. Math. Ann., 328(4):701–706, 2004. URL: https://doi-org/10.1007/ s00208-003-0505-y.

[21] S. Helgason. Diﬀerential geometry, Lie groups, and symmetric spaces, volume 80 of Pure and Applied Mathematics. Academic Press, Inc., 1978.

[22] K. A. Hirsch. On inﬁnite soluble groups. III. Proc. London Math. Soc. (2), 49:184–194, 1946. URL: https://doi-org/10.1112/plms/s2-49. 3.184.

[23] F. Hirzebruch. Chern numbers of algebraic surfaces: an example. Math. Ann., 266(3):351–356, 1984. URL: https://doi-org/10.1007/ BF01475584.

[24] L. Katzarkov and M. Ramachandran. On the universal coverings of algebraic surfaces. Ann. Sci. Ecole´ Norm. Sup. (4), 31(4):525–535, 1998. URL: https://doi-org/10.1016/S0012-9593(98)80105-5.

[25] D. Kazhdan. Some applications of the Weil representation. J. Analyse Mat., 32:235–248, 1977. URL: https://doi-org/10.1007/ bf02803582.

43 [26] F. C. Kirwan, R. Lee, and S. H. Weintraub. Quotients of the complex ball by discrete groups. Paciﬁc J. Math., 130(1):115–141, 1987. URL: http://projecteuclid.org/euclid.pjm/1102690295.

[27] J. Koll´ar. Shafarevich maps and automorphic forms. M. B. Porter Lectures. Princeton University Press, 1995. URL: https://doi-org/ 10.1515/9781400864195.

[28] D. Long and A. W. Reid. Surface subgroups and subgroup separability in 3-manifold topology. Publica¸cõesMatemáticasdo IMPA. [IMPA Mathematical Publications]. Instituto Nacional de MatemáticaPura e Aplicada (IMPA), Rio de Janeiro, 2005. 25o ColóquioBrasileiro de Matemática.[25th Brazilian Mathematics Colloquium].

[29] A. Malcev. On isomorphic matrix representations of inﬁnite groups. Rec. Math. [Mat. Sbornik] N.S., 8 (50):405–422, 1940.

[30] A. I. Malcev. Nilpotent torsion-free groups. Izvestiya Akad. Nauk. SSSR. Ser. Mat., 13:201–212, 1949.

[31] A. I. Malcev. On homomorphisms onto ﬁnite groups. Ivanov. Gos. Ped. Inst. U˘c.Zap., 18:49–60, 1958.

[32] G. D. Mostow. Generalized Picard lattices arising from half-integral conditions. Inst. Hautes Etudes´ Sci. Publ. Math., (63):91–106, 1986. URL: http://www.numdam.org/item?id=PMIHES_1986__63__91_0.

[33] G. Prasad and A. S. Rapinchuk. Computation of the metaplectic kernel. Inst. Hautes Etudes´ Sci. Publ. Math., (84):91–187, 1996. URL: http: //www.numdam.org/item?id=PMIHES_1996__84__91_0.

[34] G. Prasad and S.-K. Yeung. Fake projective planes. Invent. Math., 168(2):321–370, 2007. URL: https://doi-org/10.1007/ s00222-007-0034-5.

[35] G. Prasad and S.-K. Yeung. Addendum to “Fake projective planes” Invent. Math. 168, 321–370 (2007) [mr2289867]. Invent. Math., 182(1):213–227, 2010. URL: https://doi-org/10.1007/ s00222-010-0259-6.

[36] A. W. Reid. Proﬁnite properties of discrete groups. In Groups St Andrews 2013, volume 422 of London Math. Soc. Lecture Note Ser., pages 73–104. Cambridge Univ. Press, 2015.

44 [37] G. Shimura. Automorphic forms and the periods of abelian varieties. J. Math. Soc. Japan, 31(3):561–592, 1979. URL: https://doi-org/ 10.2969/jmsj/03130561.

[38] Y. T. Siu. The complex-analyticity of harmonic maps and the strong rigidity of compact K¨ahlermanifolds. Ann. of Math. (2), 112(1):73–111, 1980. URL: https://doi-org/10.2307/1971321.

[39] E. H. Spanier. Algebraic topology. Springer, 1995. Corrected reprint of the 1966 original.

[40] M. Stover. Volumes of Picard modular surfaces. Proc. Amer. Math. Soc., 139(9):3045–3056, 2011. URL: https://doi-org/10.1090/ S0002-9939-2011-10786-X.

[41] M. Stover. Hurwitz ball quotients. Math. Z., 278(1-2):75–91, 2014. URL: https://doi-org/10.1007/s00209-014-1306-6.

[42] M. Stover. Cusp and b1 growth for ball quotients and maps onto Z with ﬁnitely generated kernel. Indiana Univ. Math. J., 70(1):213–233, 2021.

[43] M. Stover and G. Urz´ua. Rigid surfaces arbitrarily close to the Bogomolov–Miyaoka–Yau line. arXiv:1909.00435.

[44] D. Sullivan. On the intersection ring of compact three manifolds. Topology, 14(3):275–277, 1975. URL: https://doi-org/10.1016/ 0040-9383(75)90009-9.

[45] D. Toledo. Examples of fundamental groups of compact K¨ahlermanifolds. Bull. London Math. Soc., 22(4):339–343, 1990. URL: https: //doi-org/10.1112/blms/22.4.339.

[46] D. Toledo. Projective varieties with non-residually ﬁnite fundamental group. Inst. Hautes Etudes´ Sci. Publ. Math., (77):103–119, 1993. URL: http://www.numdam.org/item?id=PMIHES_1993__77__103_0.

[47] S. Troncoso and G. Urz´ua.Savage surfaces. arXiv:1912.07378.