New insights into groups with uniform exponential growth.
problem Uniform exponential growth in hierarchically hyperbolic groups.
method Quasi-isometric characterization and new insights into group structure.
result Uniform exponential growth for hierarchically hyperbolic groups.
Proves WPD elements are exponentially generic in acylindrically hyperbolic groups.
problem Growth rates of acylindrically hyperbolic groups.
method Analyzes exponential growth rates of non-WPD and WPD elements.
result WPD elements are exponentially generic in acylindrically hyperbolic groups.
New proof shows not all Salem numbers are growth rates of Coxeter groups.
problem Identifying growth rates of Coxeter groups using Salem numbers.
method New proof using spectral radii and Coxeter transformations.
result Not every Salem number is a growth rate of hyperbolic Coxeter groups.
3D hyperbolic Coxeter groups' growth rates are Perron numbers.
problem Growth rates of 3D hyperbolic Coxeter groups.
method Combining Parry's result and \cite{Y}'s main result.
result Growth rates of 3D hyperbolic Coxeter groups are Perron numbers.
Study projection in acylindrically hyperbolic groups, proving sublinear tracking and growth bounds.
problem Projection phenomena in acylindrically hyperbolic groups.
method Analyzing shortest projections in word metrics and hyperbolic spaces.
result Sublinear tracking of shortest projections and effective growth bounds.
New infinite series of hyperbolic polytopes with special growth rates found.
problem Finding new infinite series of non-compact hyperbolic polytopes.
method Constructing infinite series of non-simple ideal hyperbolic Coxeter 4-polytopes.
result Growth rates of the constructed polytopes are Perron numbers.
On the growth rates of cofinite 3-dimensional hyperbolic Coxeter groups whose dihedral angles are of the form mπ for m=2,3,4,5,6math.GT Study growth rates of specific hyperbolic Coxeter groups with fixed dihedral angles.
problem Arithmetic properties of growth rates in hyperbolic Coxeter groups.
method Analyzing arithmetic properties of growth rates for specific Coxeter groups with fixed dihedral angles.
result Growth rates are always Perron numbers.
Extends growth properties of hyperbolic groups to their extensions.
problem Quantifying subgroup alternatives in group laws.
method Develops a framework for preserving exponential growth in extensions of hyperbolic groups.
result Automorphism groups of certain hyperbolic and Artin groups have locally uniform exponential growth.
Directly proves logarithmic systolic growth for all hyperbolic surfaces.
problem Proving logarithmic systolic growth for all hyperbolic surfaces.
method Using original Brooks/Buser-Sarnak surfaces through a direct approach.
result Directly proves logarithmic systolic growth for all hyperbolic surfaces.
New hyperbolic manifolds show exponential homology torsion growth.
problem Understanding growth of torsion in homology groups of hyperbolic manifolds.
method Constructed a family of hyperbolic manifolds with specific growth properties.
result Demonstrated that recent bound on homological torsion is asymptotically sharp.
The Unified Growth Theory is a puzzling collection of myths based on illusions created by hyperbolic distributions. Some of these myths are discussed. The examination of data shows that the three stages of growth (Malthusian Regime, Post-Malthusian Regime and Modern Growth Regime) did not exist and that Industrial Revo…
Homology growth of specific mapping tori vanishes for certain groups.
problem Homology growth of polynomially growing mapping tori in various groups.
method Proof of the cheap rebuilding property for specific groups.
result Torsion homology growth vanishes for Farber sequences in every degree.
The paper proves many hyperbolic manifolds have similar growth rates to all hyperbolic manifolds.
problem Understanding growth rates of hyperbolic manifolds.
method Proving existence of many compact hyperbolic manifolds as boundaries of other hyperbolic manifolds.
result Establishes that compact hyperbolic arithmetic n-manifolds have the same growth rate as all hyperbolic n-manifolds.
Data describing historical economic growth are analysed. Included in the analysis is the world and regional economic growth. The analysis demonstrates that historical economic growth had a natural tendency to follow hyperbolic distributions. Parameters describing hyperbolic distributions have been determined. A search …
Historical income per capita data follow hyperbolic growth patterns.
problem Economic stagnation and Malthusian traps in historical data.
method Fitting hyperbolic distributions to GDP/capita and population data.
result Income per capita growth was monotonic and without transitions.
Lower bounds for systole growth in quaternionic hyperbolic manifolds.
problem Finding lower bounds for systole growth in quaternionic hyperbolic manifolds.
method Explicit lower bound calculation for systole in congruence covers.
result Optimal lower bound for systole growth proved.
Study hyperbolic geometry to find Fibonacci numbers.
problem Investigate exponential growth and isoperimetric inequality in hyperbolic geometry.
method Combinatorial approximation of hyperbolic plane, calculations.
result Surprising link to Fibonacci numbers.
We introduce and systematically study the concept of a growth tight action. This generalizes growth tightness for word metrics as initiated by Grigorchuk and de la Harpe. Given a finitely generated, non-elementary group G acting on a G--space X, we prove that if G contains a strongly contracting eleme…
Study shows exponential growth of Laplacian determinant on random hyperbolic surfaces.
problem Understanding the behavior of Laplacian determinants on random hyperbolic surfaces.
method Investigated various models of random hyperbolic surfaces and their Laplacian determinants as genus increases.
result For all popular models, the determinant grows exponentially with a universal exponent as the genus goes to infinity.
The paper calculates the volume growth of hyperbolic surfaces with short geodesics.
problem Understanding the volume growth of hyperbolic surfaces with short geodesics.
method Introduced a function L(g) to measure the length of geodesics and computed the volume growth rate.
result The volume of surfaces with short geodesics is equal to V_g almost surely as g approaches infinity.
Study shows how volume growth affects homological torsion in 3-manifolds.
problem Understanding homological torsion growth in 3-manifolds.
method Examined abelian covers of alternating links and computed homological torsion growth explicitly.
result Explicit computation of homological torsion growth in terms of volume growth.
We give an upper bound for the growth of homology torsions of finite coverings of irreducible 3-manifolds with tori boundary in terms of hyperbolic volume.
Proves minimal growth rate for Coxeter groups in hyperbolic space.
problem Finding minimal growth rate Coxeter groups in hyperbolic space.
method Generalizes methods from cocompact case to non-cocompact groups.
result Proves minimal growth rate for Coxeter groups Γn. We prove that for a relatively hyperbolic group G there is a sequence of relatively hyperbolic proper quotients such that their growth rates converge to the growth rate of G. Under natural assumptions, the same conclusion holds for the critical exponent of a cusp-uniform action of G on a hyperbolic metric space. As a c…
New groups found in hyperbolic 4D and 5D space have minimal growth rate.
problem Finding minimal growth rates in hyperbolic Coxeter groups.
method Combinatorial properties of hyperbolic Coxeter polyhedra, partial classification results, monotonicity properties of growth rates.
result Coxeter groups G4 and G5 in H4 and H5 have the smallest growth rate. Random quotients of hyperbolic cubulated groups remain cubulated.
problem Understanding properties of random quotients of hyperbolic cubulated groups.
method Cubical small-cancellation theory, exponential growth of conjugacy classes, and hyperplane stabilizers' growth.
result Low-density random quotients of cubulated hyperbolic groups are cubulated and hyperbolic.
Groups grow faster than their subgroups, proven using special tools.
problem Growth rates of subgroups within hyperbolic groups.
method Automatic structures, Perron-Frobenius theory, growth tightness, rotating families.
result Non-elementary hyperbolic groups grow exponentially faster than their quasiconvex subgroups.
Study on hyperbolic groups, focusing on separability and splittings.
problem Coarse separability and splittings in hyperbolic groups.
method Quantitative analysis of volume growth and cut-sets, focusing on thickened spheres.
result One-ended hyperbolic groups that are not virtually surface groups are coarsely separable by a subset of subexponential growth if and only if they split over a virtually cyclic subgroup.
We give a quantification of residual finiteness for the fundamental groups of hyperbolic manifolds that admit a totally geodesic immersion to a compact, right-angled Coxeter orbifold of dimension 3 or 4. Specifically, we give explicit upper bounds on residual finiteness that are linear in terms of geodesic length. We t…
The study calculates the growth rate of reciprocal hyperbolic elements in Hecke groups.
problem Counting reciprocal hyperbolic elements in Hecke groups.
method Analyzes conjugacy classes of hyperbolic elements associated with reciprocal geodesics.
result Determines the asymptotic growth rate and limiting constant of primitive conjugacy classes of reciprocal hyperbolic elements.
New trick builds hyperbolic manifolds from compact ones, proving some don't virtually fiber.
problem Proving some hyperbolic manifolds don't virtually fiber.
method Hyperbolic reflection group trick, embedding theory, manifold topology.
result Constructed Gromov hyperbolic 7-manifolds that don't virtually fiber over a circle.
Torsion in cohomology grows exponentially for certain arithmetic hyperbolic manifolds.
problem Growth of torsion in cohomology of arithmetic hyperbolic manifolds.
method Study of sequences of arithmetic subgroups of SO_0(d,1) and Spin(d,1) yielding hyperbolic manifolds.
result Torsion in cohomology grows exponentially under natural assumptions.
Simplifies analysis of hyperbolic distributions in demographic and economic research.
problem Fundamental postulates of demographic and economic research contradicted by data.
method Simple method of reciprocal values for identifying and analyzing hyperbolic distributions.
result Fundamental postulates of demographic and economic research are incorrect.
Groups with hyperbolic properties don't have strong Property (T).
problem Proving groups with hyperbolic properties don't have strong Property (T).
method Constructing an unbounded affine representation with polynomial growth.
result Groups with hyperbolic properties do not have strong Property (T).
We prove that the semistability growth of hyperbolic groups is linear, which implies that hyperbolic groups which are sci (simply connected at infinity) have linear sci growth. Based on the linearity of the end-depth of finitely presented groups we show that the linear sci is preserved under amalgamated products over f…
Study growth rates of subgroups in groups with a constricting element.
problem Understanding growth rates of subgroups in groups with a constricting element.
method Examining the spectrum of relative and quotient exponential growth rates of quasi-convex subgroups.
result Determine when growth rates of subgroups are strictly smaller or coincide with the group's growth rate.
In [6], Kellerhals and Perren conjectured that the growth rates of the reflection groups given by hyperbolic Coxeter polyhedra are always Perron numbers. We prove that this conjecture is always true for the case of ideal Coxeter polyhedra in H3. We also find out the ideal Coxeter polyhedron in $\mathbb{H}^3…
The study of double coset growth in specific groups confirms a conjecture about generic 3-manifolds.
problem Understanding the growth of double cosets in specific groups.
method Generalizing previous work on hyperbolic groups, the study proves the growth of double cosets for certain subgroups.
result The double coset growth of certain subgroups is comparable to the orbital growth function.
The paper studies the growth of lengths of curves on hyperbolic surfaces.
problem Asymptotic growth of lengths of closed curves on hyperbolic surfaces.
method Ergodic properties of the earthquake flow on moduli space.
result Asymptotics of the number of closed curves of length ≤ L on a hyperbolic surface.
The paper studies the growth of closed geodesics on hyperbolic surface amalgams.
problem Understanding the growth of closed geodesics on hyperbolic surface amalgams.
method Analyzing topological and volume entropies, and their dependence on geometric data.
result Entropy can increase exponentially with pasting length in the absence of a lower bound on the systole.
This note addresses some questions that arise in the series of works by Kyoji Saito on the growth functions of graphs. We study "hyperbolike" graphs, which include Cayley graphs of hyperbolic groups. We generalize some well-known results on hyperbolic groups to the hyperbolike setting, including rationality of generati…
The study counts arcs on hyperbolic surfaces, providing asymptotic growth formulas.
problem Counting arcs on hyperbolic surfaces with boundaries and cusps.
method Asymptotic analysis of pure mapping class group orbits and arc lengths.
result The number of arcs of bounded length is asymptotically proportional to L6g−6+2(n+p). The paper calculates the growth rates of billiard languages in hyperbolic polygons.
problem Computing the exponential growth rates of billiard languages in polygons.
method New methods relating to minimal tiling paths.
result Explicit computation of exponential growth rates for q even, and bounds for q odd. We study the set G of growth rates of of ideal Coxeter groups in hyperbolic 3-space which consists of real algebraic integers greater than 1. We show that (1) G is unbounded above while it has the minimum, (2) any element of G is a Perron number, and (3) growth rates of of ideal Coxeter groups with n generators are l…
Random surfaces' diameter grows logarithmically with size.
problem Estimating the diameter of random hyperbolic surfaces.
method Uniform gluing of triangles, compactification, asymptotic analysis.
result The diameter is asymptotic to 2logn. Random walks on hyperbolic spaces show linear growth in translation lengths.
problem Investigate the growth of translation lengths in random walks on hyperbolic spaces.
method Prove linear growth without moment conditions and apply to Teichmüller spaces.
result Linear growth of translation lengths in random walks on hyperbolic spaces.
Disk Embeddings tackle embedding DAGs with exponential growth.
problem Embedding DAGs with exponentially increasing ancestors and descendants.
method Disk Embeddings framework for quasi-metric spaces, including Hyperbolic Disk Embeddings.
result Disk Embeddings outperform existing methods in complex DAGs.
Abstract Coxeter groups have growth rates that are Perron numbers.
problem Understanding growth rates of Coxeter groups.
method Defined a class of Coxeter groups, ∞--spanned, and analyzed their growth rates. result For ∞--spanned Coxeter groups, geodesic growth rate strictly dominates word growth rate and appears to be a Perron number.