Related papers: Pair correlation of hyperbolic lattice angles

Let $\Gamma< \mathrm{PSL}_2(\mathbb{R})$ be a lattice and $\omega\in \mathbb{H}$ a point in the upper half plane. We prove the existence and give an explicit formula for the pair correlation density function for the set of angles between…

For every two points $z_0,z_1$ in the upper half-plane, consider all elements $\gamma$ in the principal congruence group $\Gamma(N)$, acting on the upper half-plane by fractional linear transformations, such that the hyperbolic distance…

It is well known that the angles in a lattice acting on hyperbolic $n$-space become equidistributed. In this paper we determine a formula for the pair correlation density for angles in such hyperbolic lattices. Using this formula we…

For $\Gamma$ a cocompact or cofinite Fuchsian group, we study the hyperbolic lattice point problem in conjugacy classes, which is a modification of the classical hyperbolic lattice point problem. We use large sieve inequalities for the…

Let $\Gamma$ be a cocompact discrete subgroup of $\mathrm{PSL}_{2}(\mathbb{C})$ and denote by $\mathcal{H}$ the three dimensional upper half-space. For a $p\in\mathcal{H}$, we count the number of points in the orbit $\Gamma p$, according to…

We construct a Cayley graph $\mathbf{Cay}_S(\Gamma)$ of a hyperbolic group $\Gamma$ such that there are elements $g,h\in\Gamma$ and a point $\gamma \in \partial_\infty\Gamma = \partial_\infty\mathbf{Cay}_S(\Gamma)$ such that the sets…

Let \Gamma be a lattice in G=SL(n,R) and X=G/S a homogeneous space of G, where S is a closed subgroup of G which contains a real algebraic subgroup H such that G/H is compact. We establish uniform distribution of orbits of \Gamma in X…

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…

Let $\Gamma \subset \operatorname{PU}(1,n)$ be a lattice, and $S_\Gamma$ the associated ball quotient. We prove that, if $S_\Gamma$ contains infinitely many maximal totally geodesic subvarieties, then $\Gamma$ is arithmetic. We also prove…

We show that an isometric action of a torsion-free uniform lattice $\Gamma$ on hyperbolic space $\mathbb{H}^n$ can be metrically approximated by geometric actions of $\Gamma$ on $\mathrm{CAT}(0)$ cube complexes, provided that either $n$ is…

For $n\geq 3$ and $\Gamma$ a cocompact lattice acting on the hyperbolic space $\mathbb{H}^n$, we investigate the average behaviour of the error term in the circle problem. First, we explore the local average of the error term over compact…

We study the fine distribution of lattice points lying on expanding circles in the hyperbolic plane $\mathbb{H}$. The angles of lattice points arising from the orbit of the modular group $PSL_{2}(\mathbb{Z})$, and lying on hyperbolic…

Let $M$ be either the 2-sphere $\SS^2 \subset\RR^3$ or the hyperbolic plane $\HH^2 \subset \RR^3$. If $\Delta(abc)$ is a geodesic triangle on $M$ with corners at $a,b,c\in M$, we denote by $\alpha, \beta, \gamma\in M$ the midpoints of their…

Let x be a point in R^2 with irrational slope and let \Gamma denote the lattice SL(2,Z) acting linearly on R^2. Then, the orbit \Gamma x is dense in R^2. We give efective results on the approximation of a point y in R^2 by points of the…

We study the distribution of orbits of a lattice $\Gamma\leq\text{SL}(3,\mathbb R)$ in the moduli space $X_{2,3}$ of covolume one rank-two discrete subgroups in $\mathbb R^3$. Each orbit is dense, and our main result is the limiting…

The hyperbolic lattice point problem asks to estimate the size of the orbit $\Gamma z$ inside a hyperbolic disk of radius $\cosh^{-1}(X/2)$ for $\Gamma$ a discrete subgroup of $\hbox{PSL}_2(R)$. Selberg proved the estimate $O(X^{2/3})$ for…

We prove that, if a closed geodesic $\Gamma$ on a complete finite type hyperbolic surface has at least 2 self-intersections, then the length of $\Gamma$ has an lower bound $2\log(5+2\sqrt6)$, and the lower bound is sharp, attained on a…

We study the correlations of pairs of complex logarithms of $\mathbb Z$-lattice points in the complex line at various scalings, proving the existence of pair correlation functions. We prove that at the linear scaling, the pair correlations…

In this paper, we consider the geodesic flow on factors of the hyperbolic plane. We prove that a periodic orbit including a 2-antiparallel encounter has a partner orbit. We construct the partner orbit and give an estimate for the action…

Let $\Gamma$ be a geometrically-finite Fuchsian group acting on the upper half plane $\hh.$ Let $\E$ denote the set of elliptic fixed points of $\Gamma$ in $\hh.$ We give a lower bound on the minimal hyperbolic distance between points in…