Arithmetic Lattice Research Articles

Profinite non-rigidity of arithmetic groups

We show that for a typical higher-rank arithmetic lattice \Gamma , there exist finite index subgroups \Gamma_{1} and \Gamma_{2} such that \Gamma_{1}\not\cong \Gamma_{2} while \widehat{\Gamma_{1}}\cong \widehat{\Gamma_{2}} . But there are exceptions to that rule.

Arithmetic Trialitarian Hyperbolic Lattices Are Not Locally Extended Residually Finite

Abstract A group is LERF (locally extended residually finite) if all its finitely generated subgroups are separable. We prove that the trialitarian arithmetic lattices in $\mathbf{PSO}_{7,1}(\mathbb{R})$ are not LERF. This result, together with previous work by the third author, implies that no arithmetic lattice in $\mathbf{PO}_{n,1}(\mathbb{R})$, $n>3$, is LERF.

A hyperbolic lattice in each dimension with Zariski-dense surface subgroups

For each integer n ≥ 3 n \geq 3 , we exhibit a nonuniform arithmetic lattice in S O ( n , 1 ) \mathrm {SO}(n,1) containing Zariski-dense surface subgroups.

Generalization of Selberg's 3/16 theorem for convex cocompact thin subgroups of SO(n,1)

Let Γ be a convex cocompact thin subgroup of an arithmetic lattice in SO(n,1). We generalize Selberg's 316 theorem in this setting, i.e., we prove uniform exponential mixing of the frame flow and obtain a uniform resonance-free half plane for the congruence covers of the hyperbolic manifold Γ﹨Hn. This extends the work of Oh–Winter who established the n=2 case. The theorem follows from uniform spectral bounds for the congruence transfer operators with holonomy. We employ Sarkar–Winter's frame flow version of Dolgopyat's method uniformly over the congruence covers as well as Golsefidy–Varjú's generalization of Bourgain–Gamburd–Sarnak's expansion machinery by using the properties that the return trajectory subgroups are Zariski dense and have trace fields which coincide with that of Γ. These properties follow by proving that the return trajectory subgroups are of finite index in Γ.

Thin right‐angled Coxeter groups in some uniform arithmetic lattices

Using a variant of an unpublished argument due to Agol, we show that an irreducible right-angled Coxeter group on n ⩾ 3 ${n \geqslant 3}$ vertices embeds as a thin subgroup of a uniform arithmetic lattice in an indefinite orthogonal group O ( p , q ) $\mathrm{O}(p, q)$ for some p , q ⩾ 1 $p, q \geqslant 1$ satisfying p + q = n $p+q=n$ .

Translates of homogeneous measures associated with observable subgroups on some homogeneous spaces

In the present article, we study the following problem. Let $\boldsymbol {G}$ be a linear algebraic group over $\mathbb {Q}$, let $\Gamma$ be an arithmetic lattice, and let $\boldsymbol {H}$ be an observable $\mathbb {Q}$-subgroup. There is a $H$-invariant measure $\mu _H$ supported on the closed submanifold $H\Gamma /\Gamma$. Given a sequence $(g_n)$ in $G$, we study the limiting behavior of $(g_n)_*\mu _H$ under the weak-$*$ topology. In the non-divergent case, we give a rather complete classification. We further supplement this by giving a criterion of non-divergence and prove non-divergence for arbitrary sequence $(g_n)$ for certain large $\boldsymbol {H}$. We also discuss some examples and applications of our result. This work can be viewed as a natural extension of the work of Eskin–Mozes–Shah and Shapira–Zheng.

Cusp and b_1 growth for ball quotients and maps onto Z with finitely generated kernel

Let $M = \mathbb{B}^2 / \Gamma$ 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 $\Gamma \subset \mathrm{PU}(n, 1)$ of simplest type contains a finite index subgroup with this property.

Constructing thin subgroups of SL(n + 1, ℝ) via bending

In this paper we use techniques from convex projective geometry to produce many new examples of thin subgroups of lattices in special linear groups that are isomorphic to the fundamental groups of finite volume hyperbolic manifolds. More specifically, we show that for a large class of arithmetic lattices in SO(n,1) it is possible to find infinitely many non-commensurable lattices in SL(n+1,R) that contain a thin subgroup isomorphic to a finite index subgroup of the original arithmetic lattice. This class of arithmetic lattices includes all non-cocompact arithmetic lattices and all cocompact arithmetic lattices when $n$ is even.

Arithmeticity of discrete subgroups containing horospherical lattices

Let $G$ be a semisimple real algebraic Lie group of real rank at least two and $U$ be the unipotent radical of a non-trivial parabolic subgroup. We prove that a discrete Zariski dense subgroup of $G$ that contains an irreducible lattice of $U$ is an arithmetic lattice of $G$. This solves a conjecture of Margulis and extends previous work of Hee Oh.

Arithmetic lattices in unipotent algebraic groups

AbstractFixing an arithmetic lattice Γ in an algebraic groupG, the commensurability growth function assigns to eachnthe cardinality of the set of subgroups Δ with[Γ:Γ∩Δ][Δ:Γ∩Δ]=n{[\Gamma:\Gamma\cap\Delta][\Delta:\Gamma\cap\Delta]=n}. This growth function gives a new setting where methods of F. Grunewald, D. Segal and G. C. Smith’s “Subgroups of finite index in nilpotent groups” apply to study arithmetic lattices in an algebraic group. In particular, we show that, for any unipotent algebraicℤ{\mathbb{Z}}-group with arithmetic lattice Γ, the Dirichlet function associated to the commensurability growth function satisfies an Euler decomposition. Moreover, the local parts are rational functions inp-s{p^{-s}}, where the degrees of the numerator and denominator are independent ofp. This gives regularity results for the set of arithmetic lattices inG.

GEOMETRIC CYCLES IN COMPACT LOCALLY HERMITIAN SYMMETRIC SPACES AND AUTOMORPHIC REPRESENTATIONS

Let G be a linear connected non-compact real simple Lie group and let K ⊂ G be a maximal compact subgroup of G. Suppose that the centre of K is isomorphic to 𝕊1 so that X := G/K is a global Hermitian symmetric space. Denote the compact dual of X by Xu. Let θ be the Cartan involution of G that fixes K. Let Γ be a uniform arithmetic lattice in G such that θ(Γ) = Γ. Suppose that G is one of the groups SU(p, q), p < q – 1, q ≥ 5, SO0(2, q), Sp(n, ℝ), n ≠ 4, SO*(2n), n ≥ 9. Then there exists a unique irreducible unitary representation $$ {\mathcal{A}}_{\mathfrak{q}} $$ associated to a proper θ-stable parabolic subalgebra $$ \mathfrak{q} $$ with R+( $$ \mathfrak{q} $$ ) = R−( $$ \mathfrak{q} $$ ) such that if Hs,s( $$ \mathfrak{g} $$ , K; $$ {A}_{\mathfrak{q}\prime, K} $$ ) ≠ 0 for some 0 < s ≤ R+( $$ \mathfrak{q} $$ ), then $$ {\mathcal{A}}_{\mathfrak{q}\prime } $$ is unitarily equivalent to either the trivial representation or to $$ {\mathcal{A}}_{\mathfrak{q}} $$ . As a consequence, using certain geometric cycles C ⊂ XΓ := Γ\X constructed by Millson and Raghunathan, we show that under suitable hypotheses on Γ, for any sublattice Λ ⊂ Γ, the representation $$ {\mathcal{A}}_{\mathfrak{q}} $$ occurs in L2(Λ\G) with positive multiplicity.

Commensurability growths of algebraic groups

Fixing a subgroup $$\varGamma $$ in a group G, the full commensurability growth function assigns to each n the cardinality of the set of subgroups $$\varDelta $$ of G with $$[\varGamma : \varGamma \cap \varDelta ][\varDelta : \varGamma \cap \varDelta ] \le n$$. For pairs $$\varGamma \le G$$, where G is a higher rank Chevalley group scheme defined over $${\mathbb {Z}}$$ and $$\varGamma $$ is an arithmetic lattice in G, we give precise estimates for the full commensurability growth, relating it to subgroup growth and a computable invariant that depends only on G.

First order rigidity of non-uniform higher rank arithmetic groups

If $$\Gamma $$ is an irreducible non-uniform higher-rank characteristic zero arithmetic lattice (for example $${{\,\mathrm{SL}\,}}_n({\mathbb {Z}})$$ , $$n \ge 3$$ ) and $$\Lambda $$ is a finitely generated group that is elementarily equivalent to $$\Gamma $$ , then $$\Lambda $$ is isomorphic to $$\Gamma $$ .

On the linearity of lattices in affine buildings and ergodicity of the singular Cartan flow

Let X X be a locally finite irreducible affine building of dimension ≥ 2 \geq 2 , and let Γ ≤ Aut ⁡ ( X ) \Gamma \leq \operatorname {Aut}(X) be a discrete group acting cocompactly. The goal of this paper is to address the following question: When is Γ \Gamma linear? More generally, when does Γ \Gamma admit a finite-dimensional representation with infinite image over a commutative unital ring? If X X is the Bruhat–Tits building of a simple algebraic group over a local field and if Γ \Gamma is an arithmetic lattice, then Γ \Gamma is clearly linear. We prove that if X X is of type A ~ 2 \widetilde {A}_2 , then the converse holds. In particular, cocompact lattices in exotic A ~ 2 \widetilde {A}_2 -buildings are nonlinear. As an application, we obtain the first infinite family of lattices in exotic A ~ 2 \widetilde {A}_2 -buildings of arbitrarily large thickness, providing a partial answer to a question of W. Kantor from 1986. We also show that if X X is Bruhat–Tits of arbitrary type, then the linearity of Γ \Gamma implies that Γ \Gamma is virtually contained in the linear part of the automorphism group of X X ; in particular, Γ \Gamma is an arithmetic lattice. The proofs are based on the machinery of algebraic representations of ergodic systems recently developed by U. Bader and A. Furman. The implementation of that tool in the present context requires the geometric construction of a suitable ergodic Γ \Gamma -space attached to the the building X X , which we call the singular Cartan flow.

Counting points of schemes over finite rings and counting representations of arithmetic lattices

We relate the singularities of a scheme $X$ to the asymptotics of the number of points of $X$ over finite rings. This gives a partial answer to a question of Mustata. We use this result to count representations of arithmetic lattices. More precisely, if $\Gamma$ is an arithmetic lattice whose $\mathbb{Q}$-rank is greater than one, let $r_n(\Gamma)$ be the number of irreducible $n$-dimensional representations of $\Gamma$ up to isomorphism. We prove that there is a constant $C$ (for example, $C=746$ suffices) such that $r_n(\Gamma)=O(n^C)$ for every such $\Gamma$. This answers a question of Larsen and Lubotzky.

Quaternionic arithmetic lattices of rank 2 and a fake quadric in characteristic 2

We construct a torsion-free arithmetic lattice in [Formula: see text] arising from a quaternion algebra over [Formula: see text]. It is the fundamental group of a square complex with universal covering [Formula: see text], a product of trees with constant valency [Formula: see text], which has minimal Euler characteristic. Furthermore, our lattice gives rise to a fake quadric over [Formula: see text] by means of non-archimedean uniformization.

Classification and arithmeticity of toroidal compactifications with 3c2=c12= 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.

An approximation principle for congruence subgroups

The motivating question of this paper is roughly the following: given a flat group scheme G over \mathbb Z_p , p prime, with semisimple generic fiber G_{\mathbb Q_p} , how far are open subgroups of G(\mathbb Z_p) from subgroups of the form X(\mathbb Z_p)\mathbf K_p(p^n) , where X is a subgroup scheme of G and \mathbf K_p(p^n) is the principal congruence subgroup Ker (G(\mathbb Z_p)\to G(\mathbb Z/p^n\mathbb Z)) ? More precisely, we will show that for G_{\mathbb Q_p} simply connected there exist constants J\ge1 and \varepsilon&gt;0 , depending only on G , such that any open subgroup of G (\mathbb Z_p) of level p^n admits an open subgroup of index \le J which is contained in X(\mathbb Z_p)\mathbf K_p(p^{\lceil \varepsilon n\rceil}) for some proper, connected algebraic subgroup X of G defined over \mathbb Q_p . Moreover, if G is defined over \mathbb Z , then \varepsilon and J can be taken independently of p . We also give a correspondence between natural classes of \mathbb Z_p -Lie subalgebras of \mathfrak {g}_{\mathbb Z_p} and of closed subgroups of G(\mathbb Z_p) that can be regarded as a variant over \mathbb Z_p of Nori's results on the structure of finite subgroups of GL (N_0,\mathbb F_p) for large p [Nor 87]. As an application we give a bound for the volume of the intersection of a conjugacy class in the group G (\hat{\mathbb Z}) = \prod_p G (\mathbb Z_p) , for G defined over \mathbb Z , with an arbitrary open subgroup. In a companion paper, we apply this result to the limit multiplicity problem for arbitrary congruence subgroups of the arithmetic lattice G (\mathbb Z) .

Moduli Spaces of Flat Tori with Prescribed Holonomy

We generalise to the genus one case several results of Thurston concerning moduli spaces of flat Euclidean structures with conical singularities on the two dimensional sphere. More precisely, we study moduli spaces of flat tori with n cone points and a prescribed holonomy $${\rho}$$ . In his paper `Flat Surfaces’ Veech has established that under some assumptions on the cone angles, such a moduli space $${\mathcal{F}_{[\rho]} \subset \mathcal{M}_{1,n}}$$ carries a natural geometric structure modeled on the complex hyperbolic space $${{\mathbb C}{\mathbb{H}}^{n-1}}$$ which is not metrically complete. Using surgeries for flat surfaces, we prove that the metric completion $${\overline{\mathcal{F}_{[\rho]}}}$$ is obtained by adjoining to $${ \mathcal{F}_{[\rho]}}$$ certain strata that are themselves moduli spaces of flat surfaces of genus 0 or 1, obtained as degenerations of the flat tori whose moduli space is $${ \mathcal{F}_{[\rho]}}$$ . We show that the $${{\mathbb C}{\mathbb{H}}^{n-1}}$$ -structure of $${ \mathcal{F}_{[\rho]}}$$ extends to a complex hyperbolic cone-manifold structure of finite volume on $${ \overline{\mathcal{F}_{[\rho]}}}$$ and we compute the cone angles associated to the different strata of codimension 1. Finally, we address the question of whether or not the holonomy of Veech’s $${{\mathbb C}{\mathbb{H}}^{n-1}}$$ -structure on $${\mathcal{F}_{[\rho]}}$$ has a discrete image in $${ {\rm Aut}({\mathbb C}{\mathbb{H}}^{n-1})={\rm PU}(1,n-1)}$$ . We outline a general strategy to find moduli spaces $${\mathcal{F}_{[\rho]}}$$ whose $${{\mathbb C}{\mathbb{H}}^{n-1}}$$ -holonomy gives rise to lattices in $${{\rm PU}(1,n-1)}$$ and eventually we give a finite list of $${\mathcal{F}_{[\rho]}}$$ ’s whose holonomy is a complex hyperbolic arithmetic lattice.

Quantitative spectral gap for thin groups of hyperbolic isometries

Let \Lambda be a subgroup of an arithmetic lattice in \mathrm{SO}(n+1 , 1) . The quotient \mathbb{H}^{n+1} / \Lambda has a natural family of congruence covers corresponding to ideals in a ring of integers. We establish a super-strong approximation result for Zariski-dense \Lambda with some additional regularity and thickness properties. Concretely, this asserts a quantitative spectral gap for the Laplacian operators on the congruence covers. This generalizes results of Sarnak and Xue (1991) and Gamburd (2002).