arXiv research

Paper derives asymptotic formula for conjugacy classes in groups with contracting elements.

We derive a generating series for the number of free subgroups of finite index in $Δ^+ = \mathbb{Z}_p*\mathbb{Z}_q$ by using a connection between free subgroups of $Δ^+$ and certain hypermaps (also known as ribbon graphs or "fat" graphs), and show that this generating series is transcendental. We provide non-linear rec…

We show that every non-decreasing function $f\colon \mathbb N\to \mathbb N$ bounded from above by $a^n$ for some $a\ge 1$ can be realized (up to a natural equivalence) as the conjugacy growth function of a finitely generated group. We also construct a finitely generated group $G$ and a subgroup $H\le G$ of index 2 such…

Polynomial growth elements found in all subgroups of Out(F_n).

New growth rate for pseudo-Anosov conjugacy classes in Teichmüller space.

Estimates growth of reciprocal classes in Hecke groups.

Study on the number of volume-preserving Dehn fillings of hyperbolic 3-manifolds.

Given an automorphism of a free group $F_n$, we consider the following invariants: $e$ is the number of exponential strata (an upper bound for the number of different exponential growth rates of conjugacy classes); $d$ is the maximal degree of polynomial growth of conjugacy classes; $R$ is the rank of the fixed subgrou…

The study calculates the growth rate of reciprocal hyperbolic elements in Hecke groups.

The study examines the growth of conjugacy classes in negatively curved manifolds.

In this paper we derive a generating series for the number of cellular complexes known as pavings or three-dimensional maps, on $n$ darts, thus solving an analogue of Tutte's problem in dimension three. The generating series we derive also counts free subgroups of index $n$ in $Δ^+ = \mathbb{Z}_2*\mathbb{Z}_2*\mathbb{Z…

The paper finds formulas for word lengths and conjugacy classes in surface groups.

Study shows small transcendental simplicial volumes exist on manifolds.

Transcendental holomorphic Morse inequalities aim at characterizing the positivity of transcendental cohomology classes of type $(1,1)$. In this paper, we prove a weak version of Demailly's conjecture on transcendental Morse inequalities on compact Kähler manifolds. And as a consequence, we partially improve a result o…

Let X be a Hadamard manifold and $Γ$ a discrete group of isometries of X which contains an axial isometry without invariant flat half plane. We study the behavior of conformal densities on the geometric limit set of $Γ$ in order to derive a new asymptotic estimate for the growth rate of closed geodesics in not necessar…

Random quotients of hyperbolic cubulated groups remain cubulated.

Defines operations in non-Archimedean metrics theory.

For the free group $F_r$ on $r>1$ generators (respectively, the free product $G_1 * G_2$ of two nontrivial finite groups $G_1$ and $G_2$), we obtain the asymptotic for the number of conjugacy classes of commutators in $F_r$ (respectively, $G_1 * G_2$) with a given word length in a fixed set of free generators (respecti…

In this article we describe the summit sets in B_3, the smallest element in a summit set and we compute the Hilbert series corresponding to conjugacy classes.The results will be related to Birman-Menesco classification of knots with braid index three or less than three.

Study of holomorphic correspondences combining entire maps and Fuchsian groups.

Study on geodesics and dihedral groups in lattices.

Weak base-point freeness leads to Kähler-Ricci flow diameter bounds.

New proofs for growth series of Coxeter groups using complex structures.

Inspired by results of Eskin and Mirzakhani counting closed geodesics of length $\le L$ in the moduli space of a fixed closed surface, we consider a similar question in the $Out(F_r)$ setting. The Eskin-Mirzakhani result can be equivalently stated in terms of counting the number of conjugacy classes (within the mapping…

Constructs equivariant analytic torsion for proper actions on manifolds.

We derive a formula for the regularized trace of operators with compact spectrum which act on the space of square integrable functions on the quotient of a semisimple Liegroup of real rank one by a convex-cocompact subgroup. The sum of normalized orbital integrals associated to the hyperbolic conjugacy classes of this …

TAMD prevents degeneracy in finite mixtures, offering strong guarantees but modest practical improvements.

Paper introduces a new invariant for planar knotoids.

The transcendental Hodge lattice of a projective manifold $M$ is the smallest Hodge substructure in $p$-th cohomology which contains all holomorphic $p$-forms. We prove that the direct sum of all transcendental Hodge lattices has a natural algebraic structure, and compute this algebra explicitly for a hyperkahler manif…

In this work we describe a new invariant of virtual knots. We show that this transcendental function invariant generalizes several polynomial invariants of virtual knots, such as the writhe polynomial, the affine index polynomial and the zero polynomial.

Method reconstructs aneurysm growth history from patient parameters using physics-informed autoencoder.

Yair Minsky showed that punctured torus groups are classified by a pair of ending laminations (ν_-,ν_+). In this note, we show that there are ending laminations ν_+ such that for any choice of ν_-, the punctured torus group is transcendental as a subgroup of PSL_2 C.

Study shows volumes of complex classes can be represented by convex bodies.

Study groups with polynomial growth, finding structure and applications.

We construct infinite series of non-simple ideal hyperbolic Coxeter 4-polytopes whose growth rates are Perron numbers. This infinite series is the first example of such a non-compact infinite polytopal series.

Proves an index theorem for proper group actions on manifolds.

Introduces non-Archimedean metrics for pseudoeffective classes on Kähler manifolds.

The main result of this article is that if a $3$-manifold $M$ supports an Anosov flow, then the number of conjugacy classes in the fundamental group of $M$ grows exponentially fast with the length of the shortest orbit representative, hereby answering a question raised by Plante and Thurston in 1972. In fact we show th…

Growth functions of Coxeter groups and the Poincare series of Kleinian and Fuchsian singularities are $2$-tangle $q$-fractions.

Study b-divisors on Kähler manifolds linking them to currents.

The aim of this paper is to state and prove polynomial analogues of the classical Manning inequality relating the topological entropy of a geodesic flow with the growth rate of the volume of balls in the universal covering. To this aim we use two numerical conjugacy invariants, the {\em strong polynomial entropy $h_{po…

Study on braids' invariant growth and counting functions.

Let $X$ be a globally symmetric space of noncompact type, and $Γ\subset\Isom(X)$ a Schottky group of axial isometries. Then $M:=X/Γ$ is a locally symmetric Riemannian manifold of infinite volume. The goal of this note is to give an asymptotic estimate for the number of primitive closed geodesics in $M$ modulo free homo…

An element in Artin's braid group B_n is said to be periodic if some power of it lies in the center of B_n. In this paper we prove that all previously known algorithms for solving the conjugacy search problem in B_n are exponential in the braid index n for the special case of periodic braids. We overcome this difficult…

Formula derived for spherical growth series of specific groups.

We develop the Ercolani-Sinha construction of SU(2) monopoles and make this effective for (a five parameter family of centred) charge 3 monopoles. In particular we show how to solve the transcendental constraints arising on the spectral curve. For a class of symmetric curves the transcendental constraints become a numb…

Let $φ:G \to G$ be a group endomorphism where $G$ is a finitely generated group of exponential growth, and denote by $R(φ)$ the number of twisted $φ$-conjugacy classes. Fel'shtyn and Hill \cite{fel-hill} conjectured that if $φ$ is injective, then $R(φ)$ is infinite. This conjecture is true for automorphisms of non-elem…

This paper investigates the statistical properties of within-country GDP and industrial production (IP) growth rate distributions. Many empirical contributions have recently pointed out that cross-section growth rates of firms, industries and countries all follow Laplace distributions. In this work, we test whether als…