A Fuchsian group $\Gamma $ has a modular embedding if its adjoint trace field is a totally real number field and every unbounded Galois conjugate $\Gamma <^>\sigma $ comes equipped with a holomorphic (or conjugate holomorphic) map ${\phi <^>\sigma : \mathbb{B}<^>{1} \to \mathbb{B}<^>{1}}$ intertwining the actions of $\Gamma $ and $\Gamma <^>\sigma $ on the Poincar & eacute; disk $\mathbb{B}<^>{1}$. This paper provides the first cocompact nonarithmetic Fuchsian groups with a modular embedding that are not commensurable with a triangle group. The main result, proved using period domains, is that any immersed totally geodesic complex curve on a complex hyperbolic $2$-orbifold has a modular embedding. Another consequence is arithmeticity of totally geodesic curves on finite-volume complex hyperbolic surfaces that are commensurable with quotients of $\mathbb{B}<^>{1}$ by the group generated by reflections in quadrilaterals satisfying certain angle conditions.

This paper builds one-cusped complex hyperbolic 2-manifolds by an explicit geometric construction. Specifically, for each odd d >= 1 there is a smooth projective surface Z(d) with .Zd / Dc(1)(2)(Z(d)) =c(2)(Z(d))= 6d and a smooth irreducible curve E(d )on Z(d) of genus 1 so that Z(d) X E-d admits a finite volume uniformization by the unit ball B- 2 in C-2 . This produces one-cusped complex hyperbolic 2-manifolds of arbitrarily large volume. As a consequence, the 3-dimensional nilmanifold of Euler number 12d bounds geometrically for all odd d >= 1

This paper shows that the complex projective plane ℙ^2 can be realized as the underlying space for a closed hyperbolic 4-orbifold. This is the first example of a closed hyperbolic 4-orbifold whose underlying space is symplectic, which is related to the open question as to whether or not closed hyperbolic 4-manifolds can admit symplectic structures.

Let 𝒞 be the moduli space of smooth complex cubic surfaces and let π_1(𝒞) be its (orbifold) fundamental group. We prove that the “divisor subgroup” of π_1(𝒞) is characteristic. This can be interpreted as saying that the group theory of π_1(𝒞) “remembers” the divisor of nodal cubic surfaces. We deduce from this group-theoretic result and some basic complex analysis that 𝒞 has no nontrivial biholomorphic automorphisms as complex analytic orbifold.

This paper provides an iterative procedure for constructing hyperbolic Coxeter groups that virtually fiber over ℤ that is flexible enough to yield infinitely many isomorphism classes in each virtual cohomological dimension (vcd) n≥ 2. Our procedure combines results of Jankiewicz, Norin, and Wise with a generalization of a construction due to Osajda involving a new simplicial thickening process. We also give a topological argument showing that the vcd of the right-angled Coxeter groups produced by our construction increases by exactly one with each iteration, guaranteeing that our process produces examples of every vcd.

We investigate and compare applications of the Zilber-Pink conjecture and dynamical methods to rigidity problems for arithmetic real and complex hyperbolic lattices. Along the way, we obtain new general results about reconstructing a variation of Hodge structure from its typical Hodge locus that may be of independent interest. Applications to Siu's immersion problem are also discussed, the most general of which only requires the hypothesis that infinitely many closed geodesics map to proper totally geodesic subvarieties under the immersion.

For each prime $p$, this paper constructs compact complex hyperbolic $2$-manifolds with an isometric action of $\mathbb{Z} / p \mathbb{Z}$ that is not free and has only isolated fixed points. The case $p = 2$ is special, and finding general examples for $p=2$ is related to whether or not complex hyperbolic lattices are conjugacy separable on torsion.

This corrigendum points out an issue with our Angle Rigidity theorem, Theorem 4.1 in [J. Eur. Math. Soc. (JEMS) 23, 3591-3623 (2021)], in certain codimensions. This effects the proof of our main result on finiteness of maximal totally geodesic submanifolds of n-dimensional hyperbolic hybrids, Theorem 1.4 loc. cit., in codimension at least n2. Theorem 1.4 was later proved in greater generality by different methods, and thus holds in the full generality stated in our paper.

Let G be a real Lie group and Γ < G be a discrete subgroup of G. Is Γ residually finite? This paper describes known positive and negative results then poses some questions whose answers will lead to a fairly complete answer for lattices.

Fundamental groups of fake projective planes fall into fifty distinct isomorphism classes, one for each complex conjugate pair. We prove that this is not the case for their algebraic fundamental groups: there are only forty-six isomorphism classes. We show that there are four pairs of complex conjugate pairs of fake projective planes that are \operatorname{Aut}(\mathbb{C}) -equivalent and hence have mutually isomorphic algebraic fundamental groups. All other pairs of algebraic fundamental groups are shown to be distinct through explicit finite étale covers. As a by-product, this provides the first examples of commensurable but nonisomorphic lattices in a rank one semisimple Lie group that have isomorphic profinite completions.

Let $\Gamma $ be a lattice in $\mathrm{SO}_0(n, 1)$. We prove that if the associated locally symmetric space contains infinitely many maximal totally geodesic subspaces of dimension at least $2$, then $\Gamma$ is arithmetic. This answers a question of Reid for hyperbolic $n$-manifolds and, independently, McMullen for hyperbolic $3$-manifolds. We prove these results by proving a superrigidity theorem for certain representations of such lattices. The proof of our superrigidity theorem uses results on equidistribution from homogeneous dynamics, and our main result also admits a formulation in that language.

For any g_1, g_2 ≥ 0, this paper shows that there is a cocompact lattice Γ < PU(2,1) such that the ball quotient Γ\𝔹^2 is birational to a product C_1 × C_2 of smooth projective curves C_j of genus g_j. The only prior examples were ℙ^1 ×ℙ^1, due to Deligne–Mostow and rediscovered by many others, and a lesser-known product of elliptic curves whose existence follows from work of Hirzebruch. Combined with related new examples, this answers the rational variant of a question of Gromov in the positive for surfaces of Kodaira dimension κ≤ 0, namely that they admit deformations V^' such that there is a compact ball quotient Γ\𝔹^2 with a rational map Γ\𝔹^2 V^'. Often the proof gives the stronger conclusion that V^' is birational to a ball quotient orbifold. All examples are shown to be arithmetic, and even arithmeticity of Hirzebruch's example appears to be new.

We describe a criterion for a real or complex hyperbolic lattice to admit a residually finite rational solvable (RFRS) tower that consists entirely of congruence subgroups. We use this to show that certain Bianchi groups PSL2(O-d) are virtually fibered on congruence subgroups, and also exhibit the first examples of RFRS Kahler groups that are not a subgroup of a product of surface groups and abelian groups.

The Wiman–Edge pencil is a pencil of genus 6 curves for which the generic member has automorphism group the alternating group [Formula: see text]. There is a unique smooth member, the Wiman sextic, with automorphism group the symmetric group [Formula: see text]. Farb and Looijenga proved that the monodromy of the Wiman–Edge pencil is commensurable with the Hilbert modular group [Formula: see text]. In this note, we give a complete description of the monodromy by congruence conditions modulo 4 and 5. The congruence condition modulo 4 is new, and this answers a question of Farb–Looijenga. We also show that the smooth resolution of the Baily–Borel compactification of the locally symmetric manifold associated with the monodromy is a projective surface of general type. Lastly, we give new information about the image of the period map for the pencil.

We prove the existence of rigid compact complex surfaces of general type whose Chern slopes are arbitrarily close to the Bogomolov-Miyaoka-Yau bound of 3. In addition, each of these surfaces has first Betti number equal to 4.

Let $M$ be a compact 3-manifold and $\Gamma=\pi_1(M)$. Work of Thurston and Culler--Shalen established the $\mathrm{SL}_2(\mathbb{C})$ character variety $X(\Gamma)$ as fundamental tool in the study of the geometry and topology of $M$. This is particularly the case when $M$ is the exterior of a hyperbolic knot $K$ in $S^3$. The main goals of this paper are to bring to bear tools from algebraic and arithmetic geometry to understand algebraic and number theoretic properties of the so-called canonical component of $X(\Gamma)$, as well as distinguished points on the canonical component, when $\Gamma$ is a knot group. In particular, we study how the theory of quaternion Azumaya algebras can be used to obtain algebraic and arithmetic information about Dehn surgeries, and perhaps of most interest, to construct new knot invariants that lie in the Brauer groups of curves over number fields.

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).

We study residual finiteness for cyclic central extensions of cocompact arithmetic lattices Γ < PU(n, 1) of simple type. We prove that the preimage of Γ in any connected cover of PU(n, 1), in particular the universal cover, is residually finite. This follows from a more general theorem on residual finiteness of extensions whose characteristic class is contained in the span in H(Γ,Z) of the Poincaré duals to totally geodesic divisors on the ball quotient Γ\Bn. For n ≥ 4, if Γ is a congruence lattice, we prove residual finiteness of the central extension associated with any element of H(Γ,Z). Our main application is to existence of cyclic covers of ball quotients branched over totally geodesic divisors. This gives examples of smooth projective varieties admitting a metric of negative sectional curvature that are not homotopy equivalent to a locally symmetric manifold. The existence of such examples is new for all dimensions n ≥ 4.

We show that large classes of non-arithmetic hyperbolic $n$-manifolds, including the hybrids introduced by Gromov and Piatetski-Shapiro and many of their generalizations, have only finitely many finite-volume immersed totally geodesic hypersurfaces. In higher codimension, we prove finiteness for geodesic submanifolds of dimension at least $2$ that are maximal, i.e., not properly contained in a proper geodesic submanifold of the ambient $n$-manifold. The proof is a mix of structure theory for arithmetic groups, dynamics, and geometry in negative curvature.

We study residual finiteness for cyclic central extensions of cocompact arithmetic lattices Γ < PU(n, 1) simple type. We prove that the preimage of Γ in any connected cover of PU(n, 1), in particular the universal cover, is residually finite. This follows from a more general theorem on residual finiteness of extensions whose characteristic class is contained in the span in H(Γ,Z) of the Poincaré duals to totally geodesic divisors on the ball quotient Γ\Bn. For n ≥ 4, if Γ is a congruence lattice, we prove residual finiteness of the central extension associated with any element of H(Γ,Z). Our main application is to existence of cyclic covers of ball quotients branched over totally geodesic divisors. This gives examples of smooth projective varieties admitting a metric of negative sectional curvature that are not homotopy equivalent to a locally symmetric manifold. The existence of such examples is new for all dimensions n ≥ 4.