Finite Covolume Research Articles

On the commensurability of hyperbolic Coxeter groups

In this paper we study the commensurability of hyperbolic Coxeter groups of finite covolume, providing three necessary conditions for commensurability. Moreover we provide two new sets of generators for the field of definition in case of quasi-arithmetic hyperbolic Coxeter groups. This work is a concise version of chapters 4 and 5 of the author’s Ph.D. thesis Dotti (Groups of hyperbolic isometries and their commensurability. PhD thesis, Department of mathematics, University of Fribourg 2020).

Profinite Rigidity, Kleinian Groups, and the Cofinite Hopf Property

Let Γ be a nonelementary Kleinian group and H<Γ be a finitely generated, proper subgroup. We prove that if Γ has finite covolume, then the profinite completions of H and Γ are not isomorphic. If H has finite index in Γ, then there is a finite group onto which H maps but Γ does not. These results streamline the existing proofs that there exist full-sized groups that are profinitely rigid in the absolute sense. They build on a circle of ideas that can be used to distinguish among the profinite completions of subgroups of finite index in other contexts, for example, limit groups. We construct new examples of profinitely rigid groups, including the fundamental group of the hyperbolic 3-manifold Vol(3) and that of the 4-fold cyclic branched cover of the figure-eight knot. We also prove that if a lattice in PSL(2,C) is profinitely rigid, then so is its normalizer in PSL(2,C).

Hyperbolic Coxeter groups of minimal growth rates in higher dimensions

AbstractThe cusped hyperbolic n-orbifolds of minimal volume are well known for $n\leq 9$ . Their fundamental groups are related to the Coxeter n-simplex groups $\Gamma _{n}$ . In this work, we prove that $\Gamma _{n}$ has minimal growth rate among all non-cocompact Coxeter groups of finite covolume in $\textrm{Isom}\mathbb H^{n}$ . In this way, we extend previous results of Floyd for $n=2$ and of Kellerhals for $n=3$ , respectively. Our proof is a generalization of the methods developed together with Kellerhals for the cocompact case.

Nearly Fuchsian surface subgroups of finite covolume Kleinian groups

Let Gamma < PSL_2(C) be discrete, cofinite volume, and noncocompact. We prove that for all K > 1, there is a subgroup H < Gamma that is K-quasiconformally conjugate to a discrete cocompact subgroup of PSL_2(R). Along with previous work of Kahn and Markovic, this proves that every finite covolume Kleinian group has a nearly Fuchsian surface subgroup.

Parabolic vector bundles on Klein surfaces

Given a discrete subgroup Γ of finite co-volume of PGL(2,R), we define and study parabolic vector bundles on the quotient Σ of the (extended) hyperbolic plane by Γ. If Γ contains an orientation-reversing isometry, then the above is equivalent to studying real and quaternionic parabolic vector bundles on the orientation cover of degree two of Σ. We then prove that isomorphism classes of polystable real and quaternionic parabolic vector bundles are in a natural bijective correspondence with the equivalence classes of real and quaternionic unitary representations of Γ. Similar results are obtained for compact-type real parabolic vector bundles over Klein surfaces.

Persistent Hall rays for Lagrange spectra at cusps of Riemann surfaces

We study Lagrange spectra at cusps of finite area Riemann surfaces. These spectra are penetration spectra that describe the asymptotic depths of penetration of geodesics in the cusps. Their study is in particular motivated by Diophantine approximation on Fuchsian groups. In the classical case of the modular surface and classical Diophantine approximation, Hall proved in 1947 that the classical Lagrange spectrum contains a half-line, known as a Hall ray. We generalize this result to the context of Riemann surfaces with cusps and Diophantine approximation on Fuchsian groups. One can measure excursion into a cusp both with respect to a natural height function or, more generally, with respect to any proper function. We prove the existence of a Hall ray for the Lagrange spectrum of any non-cocompact, finite covolume Fuchsian group with respect to any given cusp, both when the penetration is measured by a height function induced by the imaginary part as well as by any proper function close to it with respect to the Lipschitz norm. This shows that Hall rays are stable under (Lipschitz) perturbations. As a main tool, we use the boundary expansion developed by Bowen and Series to code geodesics and produce a geometric continued fraction-like expansion and some of the ideas in Hall’s original argument. A key element in the proof of the results for proper functions is a generalization of Hall’s theorem on the sum of Cantor sets, where we consider functions which are small perturbations in the Lipschitz norm of the sum.

Lattices in amenable groups

Let $G$ be a locally compact amenable group. We say that $G$ has <i>property</i> (M) if every closed subgroup of finite covolume in $G$ is cocompact. A classical theorem of Mostow ensures that connected solvable Lie groups have property (M). We prove a no

On Poincaré series of half-integral weight

We use Poincare series of K-finite matrix coefficients of genuine integrable representations of the metaplectic cover of SL_2(ℝ) to construct a spanning set for the space of cusp forms S_m(Γ, χ), where Γ is a discrete subgroup of finite covolume in the metaplectic cover of SL_2(ℝ), χ is a character of Γ of finite order, and m is in 5/2+ℤ_{; ; ≥0}; ; . We give a result on the non-vanishing of the constructed cusp forms and compute their Petersson inner product with any f in S_m(Γ, χ). Using this last result, we construct a Poincare series Δ_{; ; Γ, k, m, ξ, χ}; ; in S_m(Γ, χ) that corresponds, in the sense of the Riesz representation theorem, to the linear functional f ↦ f^{; ; (k)}; ; (ξ) on S_m(Γ, χ), where ξ is in ℂ_{; ; ℑ(z)>0}; ; and k is in ℤ_{; ; ≥0}; ; . Under some additional conditions on Γ and χ, we provide the Fourier expansion of cusp forms Δ_{; ; Γ, k, m, ξ, χ}; ; and their expansion in a series of classical Poincare series.

On local holomorphic maps preserving invariant $(p, p)$-forms between bounded symmetric domains

Let $D, \Omega_1, ..., \Omega_m$ be irreducible bounded symmetric domains. We study local holomorphic maps from $D$ into $\Omega_1 \times... \Omega_m$ preserving the invariant $(p, p)$-forms induced from the normalized Bergman metrics up to conformal constants. We show that the local holomorphic maps extends to algebraic maps in the rank one case for any $p$ and in the rank at least two case for certain sufficiently large $p$. The total geodesy thus follows if $D=\mathbb{B}^n, \Omega_i = \mathbb{B}^{N_i}$ for any $p$ or if $D=\Omega_1 =...=\Omega_m$ with rank$(D)\geq 2$ and $p$ sufficiently large. As a consequence, the algebraic correspondence between quasi-projective varieties $D / \Gamma$ preserving invariant $(p, p)$-forms is modular, where $\Gamma$ is a torsion free, discrete, finite co-volume subgroup of Aut$(D)$. This solves partially a problem raised by Mok.

Hyperbolic geometry of the ample cone of a hyperkähler manifold

Let M be a compact hyperkahler manifold with maximal holonomy (IHS). The group $$H^2(M, {\mathbb {R}})$$ H 2 ( M , R ) is equipped with a quadratic form of signature $$(3, b_2-3)$$ ( 3 , b 2 - 3 ) , called Bogomolov–Beauville–Fujiki form. This form restricted to the rational Hodge lattice $$H^{1,1}(M,{\mathbb {Q}})$$ H 1 , 1 ( M , Q ) has signature (1, k). This gives a hyperbolic Riemannian metric on the projectivization H of the positive cone in $$H^{1,1}(M,{\mathbb {Q}})$$ H 1 , 1 ( M , Q ) . Torelli theorem implies that the Hodge monodromy group $$\varGamma $$ Γ acts on H with finite covolume, giving a hyperbolic orbifold $$X=H/\varGamma $$ X = H / Γ . We show that there are finitely many geodesic hypersurfaces, which cut X into finitely many polyhedral pieces in such a way that each of these pieces is isometric to a quotient $$P(M')/{\text {Aut}}(M')$$ P ( M ′ ) / Aut ( M ′ ) , where $$P(M')$$ P ( M ′ ) is the projectivization of the ample cone of a birational model $$M'$$ M ′ of M, and $${\text {Aut}}(M')$$ Aut ( M ′ ) the group of its holomorphic automorphisms. This is used to prove the existence of nef isotropic line bundles on a hyperkahler birational model of a simple hyperkahler manifold of Picard number at least 5 and also illustrates the fact that an IHS manifold has only finitely many birational models up to isomorphism (cf. Markman and Yoshioka in Int. Math. Res. Not. 2015(24), 13563–13574, 2015).

On commensurable hyperbolic Coxeter groups

For Coxeter groups acting non-cocompactly but with finite covolume on real hyperbolic space $$\mathbb H^n$$ , new methods are presented to distinguish them up to (wide) commensurability. We exploit these ideas and determine the commensurability classes of all hyperbolic Coxeter groups whose fundamental polyhedra are pyramids over a product of two simplices of positive dimensions.

Coxeter group in Hilbert geometry

A theorem of Tits and Vinberg allows to build an action of a Coxeter group € \Gamma on a properly convex open set \Omega of the real projective space, thanks to the data P of a polytope and reflection across its facets. We give sufficient conditions for such action to be of finite covolume, convex-cocompact or geometrically finite. We describe a hypothesis that makes those conditions necessary. Under this hypothesis, we describe the Zariski closure of € \Gamma , nd the maximal € \Gamma -invariant convex set, when there is a unique € \Gamma -invariant convex set, when the convex set \Omega  is strictly convex, when we can find a € \Gamma -invariant convex set  \Omega ' which is strictly convex.

Many examples of non-cocompact Fuchsian groups sitting in $$\hbox {PSL}_2({\mathbb {Q}})$$ PSL 2 ( Q )

We will explicitly construct many non-arithmetic Fuchsian groups while controlling some geometric properties of the action on the boundary of the hyperbolic plane. With this construction, we produce infinitely many noncommensurable non-cocompact Fuchsian groups of finite covolume sitting in \(\hbox {PSL}_2({\mathbb {Q}})\) so that the set of hyperbolic fixed points of each group will contain a given finite collection of elements in the boundary of the hyperbolic plane.

CoxIter – Computing invariants of hyperbolic Coxeter groups

CoxIter is a computer program designed to compute invariants of hyperbolic Coxeter groups. Given such a group, the program determines whether it is cocompact or of finite covolume, whether it is arithmetic in the non-cocompact case, and whether it provides the Euler characteristic and the combinatorial structure of the associated fundamental polyhedron. The aim of this paper is to present the theoretical background for the program. The source code is available online as supplementary material with the published article and on the author’s website (http://coxiter.rgug.ch).Supplementary materials are available with this article.

Pair correlation of hyperbolic lattice angles

Let ω be a point in the upper half plane, and let Γ be a discrete, finite covolume subgroup of PSL2(ℝ). We conjecture an explicit formula for the pair correlation of the angles between geodesic rays of the lattice Γω, intersected with increasingly large balls centered at ω. We prove this conjecture for Γ = PSL 2(ℤ) and ω an elliptic point.

Three techniques for obtaining algebraic circle packings

The main purpose of this article is to demonstrate three techniques for proving algebraicity statements about circle packings. We give proofs of three related theorems: (1) that every finite simple planar graph is the contact graph of a circle packing on the Riemann sphere, equivalently in the complex plane, all of whose tangency points, centers, and radii are algebraic, (2) that every flat conformal torus which admits a circle packing whose contact graph triangulates the torus has algebraic modulus, and (3) that if R is a compact Riemann surface of genus at least 2, having constant curvature -1, which admits a circle packing whose contact graph triangulates R, then R is isomorphic to the quotient of the hyperbolic plane by a subgroup of PSL_2(real algebraic numbers). The statement (1) is original, while (2) and (3) have been previously proved in the Ph.D. thesis of McCaughan. Our first proof technique is to apply Tarski's Theorem, a result from model theory, which says that if an elementary statement in the theory of real-closed fields is true over one real-closed field, then it is true over any real closed field. This technique works to prove (1) and (2). Our second proof technique is via an algebraicity result of Thurston on finite co-volume discrete subgroups of the orientation-preserving-isometry group of hyperbolic 3-space. This technique works to prove (1). Our first and second techniques had not previously been applied in this area. Our third and final technique is via a lemma in real algebraic geometry, and was previously used by McCaughan to prove (2) and (3). We show that in fact it may be used to prove (1) as well.

Quadrangulations of sphere and ball quotients

We give a classification of sphere quadrangulations satisfying a condition of non-negative curvature, following Thurston's classification of sphere triangulations under the same condition. The generic family of quadrangulations is parametrized by the points of positive square-norm of an integral Gaußian lattice in the six-dimensional complex Lorentz space. There is a subgroup of automorphisms of acting on this lattice whose orbits parametrize sphere quadrangulations in a one-to-one manner. This group acts discretely on the corresponding five-dimensional complex hyperbolic space; is of finite co-volume; its ball quotient is the moduli space of unordered 8 points on the Riemann sphere, and also appears in Picard-Terada-Deligne-Mostow list. Both Thurston's lattice and our lattice may be thought of as parametrizations of certain families of subgroups of the modular group; equivalently, of certain families of dessins. These families also parametrize points of a moduli space.

Effective equidistribution of the real part of orbits on hyperbolic surfaces

For non-cocompact Fuchsian groups with finite covolume we prove that the real part of the orbit of a point in the upper half-plane is equidistributed with an effective error term. This extends previous results by A. Good, M. Risager, and Z. Rudnick. We use the equidistribution result to generalize a theorem by F. Chamizo.

A Jørgensen–Thurston theorem for homomorphisms

In this note, we provide a description of the structure of homomorphisms from a finitely generated group to any torsion-free (3-dimensional) Kleinian group with uniformly bounded finite covolume. This is analogous to the Jorgensen-Thurston Theorem in hyperbolic geometry.

On Modular Ball-Quotient Surfaces of Kodaira Dimension One

Let be a lattice which is not co-ompact, of finite covolume with respect to the Bergman metric and acting freely on the open unit ball . Then the toroidal compactification is a projective smooth surface with elliptic compactification divisor . In this short note we discover a new class of unramifed ball quotients . We consider ball quotients with kod and . We prove that each minimal surface with finite Mordell-Weil group in the class described admits an étale covering which is a pull-back of . Here denotes the elliptic modular surface parametrizing elliptic curves with 6-torsion points which generate [6].