arXiv research

Study on complex hyperbolic bidisk isometries and their Dirichlet domains.

Let ${\bf H}_{\mathbb C}^n$ be the $n$-dimensional complex hyperbolic space and ${\rm SU}(n,1)$ be the (holomorphic) isometry group. An element $g$ in ${\rm SU}(n,1)$ is called loxodromic or hyperbolic if it has exactly two fixed points on the boundary $\partial {\bf H}_{\mathbb C}^n$. We classify ${\rm SU}(n,1)$ conju…

Shifts are not type-preserving on surface graphs.

Let $G \curvearrowright X$ be a nonelementary action by isometries of a hyperbolic group $G$ on a hyperbolic metric space $X$. We show that the set of elements of $G$ which act as loxodromic isometries of $X$ is generic. That is, for any finite generating set of $G$, the proportion of $X$--loxodromics in the ball of ra…

Discrete hyperbolic isometries proven via test maps.

The paper proves inequalities for isometries in loxodromic Kleinian groups.

Discrete subgroups of quaternionic hyperbolic isometries are proven under certain conditions.

The study classifies subgroups of outer automorphisms of free products.

New infinite-type loxodromic elements found in surface mapping classes.

In this paper, it is shown that every point in the hyperbolic 3-space is moved at a distance at least $0.5\log\left(12\cdot 3^{k-1}-3\right)$ by one of the isometries of length at most $k\geq 2$ in a 2-generator Klenian group $Γ$ which is torsion-free, not co-compact and contains no parabolic. Also some lower bounds fo…

Let $ξ$ and $η$ be two non--commuting isometries of the hyperbolic $3$--space $\mathbb{H}^3$ so that $Γ=\langleξ,η\rangle$ is a purely loxodromic free Kleinian group. For $γ\inΓ$ and $z\in\mathbb{H}^3$, let $d_γz$ denote the distance between $z$ and $γ\cdot z$. Let $z_1$ and $z_2$ be the mid-points of the shortest geod…

We study properties of generic elements of groups of isometries of hyperbolic spaces. Under general combinatorial conditions, we prove that loxodromic elements are generic (i.e. they have full density with respect to counting in balls for the word metric) and translation length grows linearly. We provide applications t…

Let $Sp(2,1)$ be the isometry group of the quaternionic hyperbolic plane ${{\bf H}_{\mathbb H}}^2$. An element $g$ in $Sp(2,1)$ is `hyperbolic' if it fixes exactly two points on the boundary of ${{\bf H}_{\mathbb H}}^2$. We classify pairs of hyperbolic elements in $Sp(2,1)$ up to conjugation. A hyperbolic element of $S…

The study characterizes loxodromes on specific rotational surfaces in 3D space.

In this paper, we prove important results concerning the loxodromes on an invariant surface in a three-dimensional Riemannian manifold, some of which generalize classical results about loxodromes on rotational surfaces in $\mathbb{R}^3$. In particular, we show how to parametrize a loxodrome on an invariant surface of $…

The paper studies timelike loxodromes on specific Lorentzian helicoidal surfaces.

The paper studies loxodromes on twisted surfaces in a specific 3D space.

Study spacelike loxodromes on helicoidal surfaces in Lorentzian n-space.

Characterizes loxodromic unit vector fields on punctured spheres.

Convex cores found for group actions on median spaces.

Loxodromic elements are pseudo-Anosov on specific graphs.

Study loxodromes and geodesics on rotational surfaces in pseudo-isotropic space.

New group with non-loxodromic Morse element found.

Generates special homeomorphisms for complex surfaces.

New loxodromic elements found in infinite-type surfaces.

Study of loxodromes on twisted surfaces in 3D space.

Study random walks on CAT(0) spaces with contracting elements, proving limit laws.

Curved loxodromes on spheres are explained and their ODE derived.

Study of two actions of mapping class groups on a graph and circle.

Flow on curves in inversive geometry converges to loxodromics.

The class of acylindrically hyperbolic groups, which are groups that admit a certain type of non-elementary action on a hyperbolic space, contains many interesting groups such as non-exceptional mapping class groups and $\operatorname{Out}(\mathbb F_n)$ for $n\geq 2$. In such a group, a generalized loxodromic element i…

The study examines groups acting loxodromically on hyperbolic graph products.

New lattice extensions of Schottky groups in hyperbolic space.

Artin-Tits groups act on a certain delta-hyperbolic complex, called the "additional length complex". For an element of the group, acting loxodromically on this complex is a property analogous to the property of being pseudo-Anosov for elements of mapping class groups. By analogy with a well-known conjecture about mappi…

In this paper, we obtain analogues of Jorgensen's inequality for non-elementary groups of isometries of quaternionic hyperbolic $n$-space generated by two elements, one of which is loxodromic. Our result gives some improvement over earlier results of Kim [10] and Markham [15]}. These results also apply to complex hyper…

In this paper, we obtain the differential equations of the space-like loxodromes on the non-degenerate canal surfaces depending on the causal characters of these canal surfaces and their meridians in Minkowski 3-space. Also we give an example by using Mathematica computer programme.

In this paper we prove that a fully irreducible outer automorphism relative to a non-exceptional free factor system acts loxodromically on the relative free factor complex as defined by Handel and Mosher. We also prove a north-south dynamic result for the action of such outer automorphisms on the closure of relative ou…

We construct an example of an isometric action of $F(a,b)$ on a $δ$-hyperbolic graph $Y$, such that this action is acylindrical, purely loxodromic, has asymptotic translation lengths of nontrivial elements of $F(a,b)$ separated away from $0$, has quasiconvex orbits in $Y$, but such that the orbit map $F(a,b)\to Y$ is n…

The present note deals with the dynamics of metric connections with vectorial torsion, as already described by E. Cartan in 1925. We show that the geodesics of metric connections with vectorial torsion defined by gradient vector fields coincide with the Levi-Civita geodesics of a conformally equivalent metric. By pullb…

The paper counts conjugacy classes of loxodromic elements in Anosov subgroups with a power saving error term.

Study mapping class groups of infinite type surfaces, classify loxodromic elements, and prove infinite-dimensional cohomology.

Extends Newton's minimal resistance problem to Riemannian surfaces.

We prove that finitely generated purely loxodromic subgroups of a right-angled Artin group $A(Γ)$ fulfill equivalent conditions that parallel characterizations of convex cocompactness in mapping class groups $\text{Mod}(S)$. In particular, such subgroups are quasiconvex in $A(Γ)$. In addition, we identify a milder cond…

In Garside groups, axes of Morse elements are strongly contracting.

Study projection in acylindrically hyperbolic groups, proving sublinear tracking and growth bounds.

The aim of this note is to give the simplest possible proof that Mapping Class Groups of closed hyperbolic surfaces are acylindrically hyperbolic, and more specifically that their curve graphs are hyperbolic and that pseudo-Anosovs act on them as loxodromic WPDs.

Local coordinates for non-singular pairs in complex and quaternionic hyperbolic groups.

The study shows that certain spacetimes are isospectrally rigid.