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Matthew Stover

Publications and source records attributed to Matthew Stover.

At least 37 records · Page 2Linked to original sources
arXiv2020

Arithmeticity, Superrigidity, and Totally Geodesic Submanifolds

Let $Γ$ 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 $Γ$ 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.

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arXiv2019

Congruence RFRS towers

We describe a criterion for a real or complex hyperbolic lattice to admit a RFRS tower that consists entirely of congruence subgroups. We use this to show that certain Bianchi groups $\mathrm{PSL}(\mathcal{O}_d)$ are virtually fibered on congruence subgroups, and also exhibit the first examples of RFRS Kähler groups that are not a subgroup of a product of surface groups and abelian groups.

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arXiv2019

Negative curves of small genus on surfaces

Let $X$ be an irreducible smooth geometrically integral projective surface over a field. In this paper we give an effective bound in terms of the Neron--Severi rank $ρ(X)$ of $X$ for the number of irreducible curves $C$ on $X$ with negative self-intersection and geometric genus less than $b_1(X)/4$, where $b_1(X)$ is the first étale Betti number of $X$. The proof involves a hyperbolic analog of the theory of spherical codes. More specifically, we relate these curves to the hyperbolic kissing number, and then prove upper and lower bounds for the hyperbolic kissing number in terms of the classical Euclidean kissing number.

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arXiv2019

Commensurability Classes of Fake Quadrics

A fake quadric is a smooth projective surface that has the same rational cohomology as a smooth quadric surface but is not biholomorphic to one. We provide an explicit classification of all irreducible fake quadrics according to the commensurability class of their fundamental group. To accomplish this task, we develop a number of new techniques that explicitly bound the arithmetic invariants of a fake quadric and more generally of an arithmetic manifold of bounded volume arising from a form of SL_2 over a number field.

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arXiv2019

New nonlinear hyperbolic groups

We construct nonlinear hyperbolic groups which are large, torsion-free, one-ended, and admit a finite $K(π,1)$. Our examples are built from superrigid cocompact rank one lattices via amalgamated free products and HNN extensions.

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arXiv2018

Lattices in $\mathrm{PU}(n,1)$ that are not profinitely rigid

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

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arXiv2018

Cusp and $b_1$ growth for ball quotients and maps onto $\mathbb{Z}$ with finitely generated kernel

Let $M = \mathbb{B}^2 / Γ$ be a smooth ball quotient of finite volume with first betti number $b_1(M)$ and let $\mathcal{E}(M) \ge 0$ be the number of cusps (i.e., topological ends) of $M$. We study the growth rates that are possible in towers of finite-sheeted coverings of $M$. In particular, $b_1$ and $\mathcal{E}$ have little to do with one another, in contrast with the well-understood cases of hyperbolic $2$- and $3$-manifolds. We also discuss growth of $b_1$ for congruence arithmetic lattices acting on $\mathbb{B}^2$ and $\mathbb{B}^3$. Along the way, we provide an explicit example of a lattice in $\mathrm{PU}(2, 1)$ admitting a homomorphism onto $\mathbb{Z}$ with finitely generated kernel. Moreover, we show that any cocompact arithmetic lattice $Γ\subset \mathrm{PU}(n, 1)$ of simplest type contains a finite index subgroup with this property.

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arXiv2018

Punctured spheres in complex hyperbolic surfaces and bielliptic ball quotient compactifications

In this paper, we study punctured spheres in two dimensional ball quotient compactifications $(X, D)$. For example, we show that smooth toroidal compactifications of ball quotients cannot contain properly holomorphically embedded $3$-punctured spheres. We also use totally geodesic punctured spheres to prove ampleness of $K_X + αD$ for $α\in (\frac{1}{4}, 1)$, giving a sharp version of a theorem of the first author with G. Di Cerbo. Finally, we produce the first examples of bielliptic ball quotient compactifications modeled on the Gaussian integers.

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arXiv2018

On general type surfaces with $q=1$ and $c_2 = 3 p_g$

Let $S$ be a minimal surface of general type with irregularity $q(S) = 1$. Well-known inequalities between characteristic numbers imply that $3 p_g(S) \le c_2(S) \le 10 p_g(S)$, where $p_g(S)$ is the geometric genus and $c_2(S)$ the topological Euler characteristic. Surfaces achieving equality for the upper bound are classified, starting with work of Debarre. We study equality in the lower bound, showing that for each $n \ge 1$ there exists a surface with $q = 1$, $p_g = n$, and $c_2 = 3n$. The moduli space $\mathcal{M}_n$ of such surfaces is a finite set of points, and we prove that $\#\mathcal{M}_n \to \infty$ as $n \to \infty$. Equivalently, this paper studies the number of closed complex hyperbolic $2$-manifolds of first betti number $2$ as a function of volume; in particular, such a manifold exists for every possible volume.

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arXiv2018

Finiteness of Maximal Geodesic Submanifolds in Hyperbolic Hybrids

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.

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arXiv2017

Explicit bounds from the Alon-Boppana theorem

The purpose of this paper is to give explicit methods for bounding the number of vertices of finite $k$-regular graphs with given second eigenvalue. Let $X$ be a finite $k$-regular graph and $μ_1(X)$ the second largest eigenvalue of its adjacency matrix. It follows from the well-known Alon-Boppana Theorem, that for any $ε> 0$ there are only finitely many such $X$ with $μ_1(X) < (2 - ε) \sqrt{k - 1}$, and we effectively implement Serre's quantitative version of this result. For any $k$ and $ε$, this gives an explicit upper bound on the number of vertices in a $k$-regular graph with $μ_1(X) < (2 - ε) \sqrt{k - 1}$.

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arXiv2017

Amalgam Anosov representations

Let $Γ$ be a one-ended, torsion-free hyperbolic group and let $G$ be a semisimple Lie group with finite center. Using the canonical JSJ splitting due to Sela, we define amalgam Anosov representations of $Γ$ into $G$ and prove that they form a domain of discontinuity for the action of $\mathrm{Out}(Γ)$. In the appendix, we prove, using projective Anosov Schottky groups, that if the restriction of the representation to every Fuchsian or rigid vertex group of the JSJ splitting of $Γ$ is Anosov, with respect to a fixed pair of opposite parabolic subgroups, then $ρ$ is amalgam Anosov.

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arXiv2017

Classification and arithmeticity of toroidal compactifications with $3\bar{c}_{2}=\bar{c}^{2}_{1}=3$

We classify the minimum volume smooth complex hyperbolic surfaces that admit smooth toroidal compactifications, and we explicitly construct their compactifications. There are five such surfaces and they are all arithmetic, i.e., they are associated with quotients of the ball by an arithmetic lattice. Moreover, the associated lattices are all commensurable. The first compactification, originally discovered by Hirzebruch, is the blowup of an Abelian surface at one point. The others are bielliptic surfaces blown up at one point. These appear to be the first known examples of smooth toroidal compactifications birational to bielliptic surfaces.

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arXiv2016

Constructing Geometrically Equivalent Hyperbolic Orbifolds

In this paper, we construct families of nonisometric hyperbolic orbifolds that contain the same isometry classes of nonflat totally geodesic subspaces. The main tool is a variant of the well-known Sunada method for constructing length-isospectral Riemannian manifolds that handles totally geodesic submanifolds of multiple codimensions simultaneously.

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arXiv2016

Parametrizing Shimura subvarieties of $\mathrm{A}_1$ Shimura varieties and related geometric problems

This paper gives a complete parametrization of the commensurability classes of totally geodesic subspaces of irreducible arithmetic quotients of $X_{a, b} = (\mathbf{H}^2)^a \times (\mathbf{H}^3)^b$. A special case describes all Shimura subvarieties of type $\mathrm{A}_1$ Shimura varieties. We produce, for any $n\geq 1$, examples of manifolds/Shimura varieties with precisely $n$ commensurability classes of totally geodesic submanifolds/Shimura subvarieties. This is in stark contrast with the previously studied cases of arithmetic hyperbolic $3$-manifolds and quaternionic Shimura surfaces, where the presence of one commensurability class of geodesic submanifolds implies the existence of infinitely many classes.

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arXiv2016

Bielliptic ball quotient compactifications and lattices in PU(2, 1) with finitely generated commutator subgroup

We construct two infinite families of ball quotient compactifications birational to bielliptic surfaces. For each family, the volume spectrum of the associated noncompact finite volume ball quotient surfaces is the set of all positive integral multiples of $\frac{8}{3}π^{2}$, i.e., they attain all possible volumes of complex hyperbolic $2$-manifolds. The surfaces in one of the two families have all $2$-cusps, so that we can saturate the entire volume spectrum with $2$-cusped manifolds. Finally, we show that the associated neat lattices have infinite abelianization and finitely generated commutator subgroup. These appear to be the first known nonuniform lattices in $\mathrm{PU}(2,1)$, and the first infinite tower, with this property.

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arXiv2016

Character varieties and actions on products of trees

It is well known that surface groups admit free and proper actions on finite products of infinite valence trees. In this note, we address the question of whether there can be a free and proper action on a finite product of bounded valence trees. We provide some obstructions and an arithmetic criterion for existence. The bulk of the paper is devoted to an approach to verifying the arithmetic criterion by studying the character variety of certain surface groups over fields of positive characteristic. The methods may be useful for attempting to determine when groups admit good linear representations in other contexts.

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