Zariski Dense Subgroups Research Articles

Non-concentration property of Patterson–Sullivan measures for Anosov subgroups

Abstract Let G be a connected semisimple real algebraic group. For a Zariski dense Anosov subgroup $\Gamma <G$ with respect to a parabolic subgroup $P_\theta $ , we prove that any $\Gamma $ -Patterson–Sullivan measure charges no mass on any proper subvariety of $G/P_\theta $ . More generally, we prove that for a Zariski dense $\theta $ -transverse subgroup $\Gamma <G$ , any $(\Gamma , \psi )$ -Patterson–Sullivan measure charges no mass on any proper subvariety of $G/P_\theta $ , provided the $\psi $ -Poincaré series of $\Gamma $ diverges at its abscissa of convergence. In particular, our result also applies to relatively Anosov subgroups.

Zariski dense surface subgroups in with odd

AbstractFor odd n we construct a path $\rho\;:\;\thinspace \Pi_1(S) \to SL(n\mathbb{R})$ of discrete, faithful, and Zariski dense representations of a surface group such that $\rho_t(\Pi_1(S)) \subset SL(n,\mathbb{Q})$ for every $t\in \mathbb{Q}$ .

On denseness of horospheres in higher rank homogeneous spaces

Abstract Let $ G $ be a connected semisimple real algebraic group and $\Gamma <G$ be a Zariski dense discrete subgroup. Let N denote a maximal horospherical subgroup of G, and $P=MAN$ the minimal parabolic subgroup which is the normalizer of N. Let $\mathcal E$ denote the unique P-minimal subset of $\Gamma \backslash G$ and let $\mathcal E_0$ be a $P^\circ $ -minimal subset. We consider a notion of a horospherical limit point in the Furstenberg boundary $ G/P $ and show that the following are equivalent for any $[g]\in \mathcal E_0$ : (1) $gP\in G/P$ is a horospherical limit point; (2) $[g]NM$ is dense in $\mathcal E$ ; (3) $[g]N$ is dense in $\mathcal E_0$ . The equivalence of items (1) and (2) is due to Dal’bo in the rank one case. We also show that unlike convex cocompact groups of rank one Lie groups, the $NM$ -minimality of $\mathcal E$ does not hold in a general Anosov homogeneous space.

Simple algebraic groups with the same maximal tori, weakly commensurable Zariski-dense subgroups, and good reduction

We provide a new condition for an absolutely almost simple algebraic group to have good reduction with respect to a discrete valuation of the base field which is formulated in terms of the existence of maximal tori with special properties. This characterization, in particular, shows that the Finiteness Conjecture for forms of an absolutely almost simple algebraic group over a finitely generated field that have good reduction at a divisorial set of places of the field (cf. [66]) would imply the finiteness of the genus of the group at hand. It also leads to a new phenomenon that we refer to as “killing the genus by a purely transcendental extension.” Yet another application deals with the investigation of “eigenvalue rigidity” of Zariski-dense subgroups (cf. [64]), which in turn is related to the analysis of length-commensurable Riemann surfaces and general locally symmetric spaces. Finally, we analyze the Finiteness Conjecture and the genus problem for simple algebraic groups of type F4.

Dichotomy and Measures on Limit Sets of Anosov Groups

Abstract Let $G$ be a connected semisimple real algebraic group. For a Zariski dense Anosov subgroup $\Gamma <G$, we show that a $\Gamma $-conformal measure is supported on the limit set of $\Gamma $ if and only if its dimension is $\Gamma $-critical. This implies the uniqueness of a $\Gamma $-conformal measure for each critical dimension, answering the question posed in our earlier paper with Edwards [13]. We obtain this by proving a higher rank analogue of the Hopf–Tsuji–Sullivan dichotomy for the maximal diagonal action. Other applications include an analogue of the Ahlfors measure conjecture for Anosov subgroups.

Anosov groups: local mixing, counting and equidistribution

Let $G$ be a connected semisimple real algebraic group, and $\Gamma<G$ be a Zariski dense Anosov subgroup with respect to a minimal parabolic subgroup. We describe the asymptotic behavior of matrix coefficients $\langle (\exp tv). f_1, f_2\rangle$ in $L^2(\Gamma\backslash G)$ as $t\to \infty$ for any $f_1, f_2\in C_c(\Gamma\backslash G)$ and any vector $v$ in the interior of the limit cone of $\Gamma$. These asymptotics involve higher rank analogues of Burger-Roblin measures which are introduced in this paper. As an application, for any affine symmetric subgroup $H$ of $G$, we obtain a bisector counting result for $\Gamma$-orbits with respect to the corresponding generalized Cartan decomposition of $G$. Moreover, we obtain analogues of the results of Duke-Rudnick-Sarnak and Eskin-McMullen for counting discrete $\Gamma$-orbits in affine symmetric spaces $H\backslash G$.

Zariski density and computing with S-integral groups

We generalize our methodology for computing with Zariski dense subgroups of SL(n,Z) and Sp(n,Z), to accommodate input dense subgroups H of SL(n,Q) and Sp(n,Q). A key task, backgrounded by the Strong Approximation theorem, is computing a minimal congruence overgroup of H. Once we have this overgroup, we may describe all congruence quotients of H. The case n=2 receives particular attention.

On the density of S-adic integers near some projective G-varieties

We provide some general conditions which ensure that a system of inequalities involving homogeneous polynomials with coefficients in a \(S\)-adic field has nontrivial \(S\)-integral solutions. The proofs are based on the strong approximation property for Zariski-dense subgroups and adelic geometry of numbers. We give some examples of applications for systems involving quadratic and linear forms.&#x0D;

Local Mixing of One-Parameter Diagonal Flows on Anosov Homogeneous Spaces

Abstract Let $G$ be a connected semisimple real algebraic group and $\Gamma &amp;lt; G$ be a Zariski dense Anosov subgroup with respect to a minimal parabolic subgroup. We prove local mixing of the one-parameter diagonal flow $\{\exp (t\mathsf {v}): t \in {\mathbb {R}}\}$ on $\Gamma \backslash G$ for any interior direction $\mathsf {v}$ of the limit cone of $\Gamma $ with respect to the Bowen–Margulis–Sullivan measure associated to $\mathsf {v}$. More generally, we allow a class of deviations to this flow along a direction $\mathsf {u}$ in some fixed subspace transverse to $\mathsf {v}$. We also obtain a uniform bound for the correlation function, which decays exponentially in $\|\mathsf {u}\|^2$. The precise form of the result is required for several applications such as the asymptotic formula for the decay of matrix coefficients in $L^2(\Gamma \backslash G)$ proved by Edwards–Lee–Oh.

Some asymptotic properties of random walks on homogeneous spaces

Let $ G $ be a connected semisimple real Lie group with finite center, and $ \mu $ a probability measure on $ G $ whose support generates a Zariski-dense subgroup of $ G $. We consider the right $ \mu $-random walk on $ G $ and show that each random trajectory spends most of its time at bounded distance of a well-chosen Weyl chamber. We infer that if $ G $ has rank one, and $ \mu $ has a finite first moment, then for any discrete subgroup $ \Lambda\subseteq G $, the $ \mu $-walk and the geodesic flow on $ \Lambda \backslash G $ are either both transient, or both recurrent and ergodic, thus extending a well known theorem due to Hopf–Tsuji–Sullivan–Kaimanovich dealing with the Brownian motion.

Invariant Measures for Horospherical Actions and Anosov Groups

AbstractLet $\Gamma $ be a Zariski dense Anosov subgroup of a connected semisimple real algebraic group $G$. For a maximal horospherical subgroup $N$ of $G$, we show that the space of all non-trivial $NM$-invariant ergodic and $A$-quasi-invariant Radon measures on $\Gamma \backslash G$, up to proportionality, is homeomorphic to ${\mathbb {R}}^{\text {rank}\,G-1}$, where $A$ is a maximal real split torus and $M$ is a maximal compact subgroup that normalizes $N$. One of the main ingredients is to establish the $NM$-ergodicity of all Burger–Roblin measures.

Uniqueness of Conformal Measures and Local Mixing for Anosov Groups

In the late seventies, Sullivan showed that, for a convex cocompact subgroup Γ of SO∘(n,1) with critical exponent δ>0, any Γ-conformal measure on ∂Hn of dimension δ is necessarily supported on the limit set Λ and that the conformal measure of dimension δ exists uniquely. We prove an analogue of this theorem for any Zariski dense Anosov subgroup Γ of a connected semisimple real algebraic group G of rank at most 3. We also obtain the local mixing for generalized BMS measures on Γ∖G including Haar measures.

Hilbert’s Irreducibility Theorem via Random Walks

Abstract Let $G$ be a connected linear algebraic group over a number field $K$, let $\Gamma $ be a finitely generated Zariski dense subgroup of $G(K)$, and let $Z\subseteq G(K)$ be a thin set, in the sense of Serre. We prove that, if $G/\textrm {R}_{u}(G)$ is either trivial or semisimple and $Z$ satisfies certain necessary conditions, then a long random walk on a Cayley graph of $\Gamma $ hits elements of $Z$ with negligible probability. We deduce corollaries to Galois covers, characteristic polynomials, and fixed points in group actions. We also prove analogous results in the case where $K$ is a global function field.

Spectral gap and exponential mixing on geometrically finite hyperbolic manifolds

Let M=Γ∖Hd+1 be a geometrically finite hyperbolic manifold with critical exponent exceeding d∕2. We obtain a precise asymptotic expansion of the matrix coefficients for the geodesic flow in L2(T1(M)), with exponential error term essentially as good as the one given by the spectral gap for the Laplace operator on L2(M) due to Lax and Phillips. Combined with the work of Bourgain, Gamburd, and Sarnak and its generalization by Golsefidy and Varjú on expanders, this implies uniform exponential mixing for congruence covers of M when Γ is a Zariski-dense subgroup contained in an arithmetic subgroup of SO∘(d,1).

A Density Theorem for the Difference Galois Groups of Regular Singular Mahler Equations

Abstract The difference Galois theory of Mahler equations is an active research area. The present paper aims at developing the analytic aspects of this theory. We first attach a pair of connection matrices to any regular singular Mahler equation. We then show that these connection matrices can be used to produce a Zariski-dense subgroup of the difference Galois group of any regular singular Mahler equation.

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.

ON DISCRETENESS OF SUBGROUPS OF QUATERNIONIC HYPERBOLIC ISOMETRIES

Let $\mathbf{H}_{\mathbb{H}}^{n}$ denote the $n$-dimensional quaternionic hyperbolic space. The linear group $\text{Sp}(n,1)$ acts on $\mathbf{H}_{\mathbb{H}}^{n}$ by isometries. A subgroup $G$ of $\text{Sp}(n,1)$ is called Zariski dense if it neither fixes a point on $\mathbf{H}_{\mathbb{H}}^{n}\cup \unicode[STIX]{x2202}\mathbf{H}_{\mathbb{H}}^{n}$ nor preserves a totally geodesic subspace of $\mathbf{H}_{\mathbb{H}}^{n}$. We prove that a Zariski dense subgroup $G$ of $\text{Sp}(n,1)$ is discrete if for every loxodromic element $g\in G$ the two-generator subgroup $\langle f,gfg^{-1}\rangle$ is discrete, where the generator $f\in \text{Sp}(n,1)$ is a certain fixed element not necessarily from $G$.

2-Generation of simple Lie algebras and free dense subgroups of algebraic groups

We construct generating pairs of simple Lie algebras in characteristic zero. We apply this construction to exhibit infinite series of 2-generator Zariski dense subgroups that are free of rank 2 of the simple algebraic groups SL(n,C), Sp(n,C), G2(C).

Multiplication of weak equivalence classes may be discontinuous

For a countably infinite group Γ \Gamma , let W Γ {\mathcal {W}}_{\Gamma } denote the space of all weak equivalence classes of measure-preserving actions of Γ {\Gamma } on atomless standard probability spaces, equipped with the compact metrizable topology introduced by Abért and Elek. There is a natural multiplication operation on W Γ {\mathcal {W}}_{\Gamma } (induced by taking products of actions) that makes W Γ {\mathcal {W}}_{\Gamma } an Abelian semigroup. Burton, Kechris, and Tamuz showed that if Γ {\Gamma } is amenable, then W Γ {\mathcal {W}}_{\Gamma } is a topological semigroup; i.e., the product map W Γ × W Γ → W Γ : ( a , b ) ↦ a × b {\mathcal {W}}_{\Gamma } \times {\mathcal {W}}_{\Gamma } \to {\mathcal {W}}_{\Gamma } \colon (\mathfrak {a}, \mathfrak {b}) \mapsto \mathfrak {a} \times \mathfrak {b} is continuous. In contrast to that, we prove that if Γ {\Gamma } is a Zariski dense subgroup of S L d ( Z ) {\mathrm {SL}}_d({\mathbb {Z}}) for some d ⩾ 2 d \geqslant 2 (for instance, if Γ {\Gamma } is a non-Abelian free group), then multiplication on W Γ {\mathcal {W}}_{\Gamma } is discontinuous, even when restricted to the subspace F W Γ {\mathcal {FW}}_{\Gamma } of all free weak equivalence classes.

Local-global principles in circle packings

We generalize work by Bourgain and Kontorovich [On the local-global conjecture for integral Apollonian gaskets, Invent. Math.196(2014), 589–650] and Zhang [On the local-global principle for integral Apollonian 3-circle packings, J. Reine Angew. Math.737, (2018), 71–110], proving an almost local-to-global property for the curvatures of certain circle packings, to a large class of Kleinian groups. Specifically, we associate in a natural way an infinite family of integral packings of circles to any Kleinian group${\mathcal{A}}\leqslant \text{PSL}_{2}(K)$satisfying certain conditions, where$K$is an imaginary quadratic field, and show that the curvatures of the circles in any such packing satisfy an almost local-to-global principle. A key ingredient in the proof is that${\mathcal{A}}$possesses a spectral gap property, which we prove for any infinite-covolume, geometrically finite, Zariski dense Kleinian group in$\operatorname{PSL}_{2}({\mathcal{O}}_{K})$containing a Zariski dense subgroup of$\operatorname{PSL}_{2}(\mathbb{Z})$.